Abstract
We consider the initial-value problem \(\tilde{u}_t=\Delta _x \tilde{u}(x,t)\), \(\tilde{u}(x,0)=u(x)\), where \(x\in \mathbb {R}^{n-1},t\in (0,T)\) and \(u\) belongs to certain weighted Orlicz–Slobodetskii space \(Y^{ \Phi ,\Phi }_{log}(\mathbb {R}^{n-1})\) subordinated to the logarithmic weight. We prove that under certain assumptions on Orlicz function \({\Phi }\), the solution \(\tilde{u}\) belongs to Orlicz–Sobolev space \(W^{1,{\Psi }}(\Omega \times (0,T))\) for certain function \(\Psi \) which in general dominates \(\Phi \). The typical representants are \(\Phi (\lambda )= \lambda (\mathrm{log} (2+\lambda ))^\alpha \), \(\Psi (\lambda )= \lambda (\mathrm{log} (2+\lambda ))^{\alpha +1}\) where \(\alpha > 0\).
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1 Introduction
The purpose of this paper is to study the initial-value problem for the heat equation:
where the initial function \(u\) lies in the completion of Lipschitz functions in certain weighted Orlicz–Slobodetski type space denoted by \(Y^{{\Phi },{\Phi }}_{log}(\Omega )\), \({\Phi }\) is a \(N\)-function. It consists of all \(v\in L^{\Phi }(\Omega )\) (the Orlicz space generated by \({\Phi }\)), for which the seminorm
is finite.
As our main result formulated in Theorem 7.2, we prove that if \(\Phi \) satisfies certain assumptions (Assumption B in Definition 6.1), then the solution \(\tilde{u}\) of (1.1) lies in the Orlicz–Sobolev space \(W^{1,\Psi }(\mathbb {R}^{n-1}\times (0,T))\), i. e. \(\tilde{u}\), together with its all first order partial derivatives belongs to \(L^{\Psi } (\mathbb {R}^{n-1}\times (0,T))\), where \(\Psi \) is in a sense conjugate to \(\Phi \) (see Definition 2.1). The natural representative pair of admitted functions would be functions generating the logarithmic Zygmund spaces: \(\Phi (\lambda )=\lambda ( \ln (2+\lambda ))^\alpha \) and \(\Psi (\lambda )=\lambda ( \ln (2+\lambda ))^{\alpha +1}\), where \(\alpha >0\). They cannot grow to fast. The conjugate of \(\Phi \) is equivalent to \(\exp t^{1/\alpha }\) near infinity and does not satisfy the \(\Delta _2\)-condition. Logarithmic Orlicz spaces are of particular interest in functional analysis, see, e.g. [4, 14, 19, 21].
For regularity results dealing with the initial data in the classical Besov spaces \(B_p^{\alpha ,q}(\Omega )\) we refer to papers: [20, 43, 51, 58–60] and to their references.
Our motivation to ask about regularity in the Orlicz setting comes from the fact that many mathematical models in the nonlinear elliptic and parabolic PDEs arising from the mathematical physics seem to have a good interpretation only when stated in Orlicz framework, see e.g. [3, 6, 15, 16, 37, 54]. Moreover, not much is known about regularity established for the heat equation with initial data in the Orlicz–Slobodetskii-type spaces, even in the nonweighted ones, where \(N\)-function \(\Phi \) is essentially different than \(\lambda ^p\). We focus on the paper [30] for an approach with an initial condition in the Orlicz space. For another result in this direction we refer to our recent paper [24], where we have proven that if Orlicz function \({R}\) satisfies certain assumptions (Assumption B from Definition 6.1), then the solution \(\tilde{u}\) of (1.1) lies in the Orlicz–Sobolev space \(W^{1,{R}}(\Omega \times (0,T))\). The difference between an approach presented here and our previous one is that now we provide the estimates between Orlicz–Sobolev-type spaces and Orlicz–Slobodetskii-type spaces defined by the possibly different Orlicz functions \(\Phi \) and \(\Psi \). This motivated us to consider weighted Orlicz–Slobodetskii setting, while our previous analysis did not require weights.
Weighted Sobolev–Slobodetskii spaces have appeared in many issues, stating from classical literature mostly by the Russian school: [42] (which appeared before fundamental paper by Slobodetskii [50]), [31, Section 9] [35, 39, 47, 56, 57]. Those works were involving measures being the power of the distance from the boundary: \(\mathrm{dist}(x,\partial \Omega )^\alpha \) See also books [32, 38, 53] and later works [1, 22, 26, 36, 40, 41, 46, 55].
Weighted Orlicz–Slobodetskii spaces involving general weights have appeared in old papers by Lacroix [34] and Palmieri [45]. Recently first author and Dhara [11, 12] were investigating properties of the extension operator from Orlicz–Slobodetski type space to Orlicz–Sobolev space in the weighted setting. See also [7, 8, 29, 32, 38], for some interesting related results.
Except the standard arguments based on Young and Jensen’s inequalities and the a priori estimates, we propose an approach based on obtained here pointwise estimates for the time-maximal functions of \(\tilde{u}\) and its first order derivatives (see Lemmas 4.4, 5.3 and 6.4) and Stein type theorem due to Kita, see Theorem 4.1 obtained in [28].
The estimates for solutions of the heat equation are of interest to many mathematicians form various branches of mathematics including the probability theory and analysis on metric spaces, see e.g. [2, 5, 10, 13, 17, 18, 44, 48], and the references therein.
2 Notation and preliminaries
2.1 Basic notation
Let \(\Omega \subset {\mathbb R}^n\) be an open set. By \(C^\infty (\bar{\Omega })\) we mean set of functions which have smooth extension to certain open neighborhood of \(\bar{\Omega }\). If \(f\) is defined on a set \(A\subseteq {\mathbb R}^n\), by \(f\chi _A\) we mean the function \(f\) extended by \(0\) outside \(A\). Let \(Mh(t_0):= \sup _{t>0} \frac{1}{2r}\int _{(t_0-r,t_0+r)} h(t)\,dt\) be the Hardy–Littlewood maximal function of \(h\in L^1_{\textit{loc}}({\mathbb R})\) [52]. We will be also dealing with the time-directional maximal function of function \(w\in L^1_{\textit{loc}}({\mathbb R}^{n-1}\times [0,\infty ) )\), the function
(defined for almost every \((x,t_0)\)). Having to norms \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _1\) defined on a Banach space \(X\), we will write \(\Vert \cdot \Vert \sim \Vert \cdot \Vert _1\) if norm \(\Vert \cdot \Vert \) is equivalent to \(\Vert \cdot \Vert _1\) on \(X\). Having two functions \(\Phi ,\Psi \) defined on \([0,\infty )\) we will say that \(\Psi \) dominates \(\Phi \) (\(\Phi \prec \Psi \)) if there exist constants \(C_1,C_2>0\) such that \(\Phi (x)\le C_1\Psi (C_2x)\) for every \(x>0\). Functions \(\Phi ,\Psi \) are called equivalent if \(\Psi \prec \Phi \) and \(\Phi \prec \Psi \).
The notation “\(\lesssim \)” will be used in usual manner, namely, if \(\Phi ,\Psi : \mathcal{A}\rightarrow {\mathbb R}\) are given functions, where \(\mathcal{A}\) is some abstract domain (it can be either a subset of Euclidean space, as well as a set of functions), we will write that \(\Phi \lesssim \Psi \) if there is a constant \(C>0\) such that \(\Phi (a)\le C\Psi (a)\), for every \(a\in \mathcal{A}\). When \(n\in \mathbf{N}\), we denote: \(Q'=[0,1]^{n-1}\), \(Q=[0,1]^n=Q'\times (0,1)\). By \(Lip(\Omega )\) we denote space of Lipschitz functions defined on the set \(\Omega \subseteq \mathbb {R}^n\), while by \(Lip_0(\Omega )\) we denote those elements of \(Lip(\Omega )\) which have compact support in \(\Omega \). If \(x=(x_1,\ldots ,x_n)\in \mathbb R^n\), then \(x'\) will stand for \((x_1,\ldots ,x_{n-1})\in {\mathbb {R}}^{n-1}\). By \(\ln x\) we denote the natural logarithm of a positive number \(x\).
2.2 Orlicz, Orlicz–Sobolev and Orlicz–Slobodetskii spaces
2.2.1 Orlicz space
When \(\Psi :[0,\infty )\rightarrow [0,\infty )\) is a nondecreasing convex continuous function such that \(\Psi (0)=0\) and \(\lim _{t\rightarrow \infty }\Psi (t)=+\infty \), the space
is called Orlicz space (see e.g. [49]). It is a Banach space equipped with the Luxemburg norm:
As is well known, when \(\Psi (\lambda )=\lambda ^p\) and \(p\ge 1\), then \(L^\Psi (\Omega )= L^p(\Omega )\) is the usual Lebesgue space. The same notation will be used for vector functions, \(u:\Omega \rightarrow {\mathbb R}^m\), with the formal difference that instead of \(|u(x)|\) we shall work with the Euclidean norm of the vector \(u(x)\).
We shall write that \(\Psi \in \Delta _2\) if it satisfies the \(\Delta _2\)-condition: \(\Psi (2\lambda )\le C\Psi (\lambda )\), for every \(\lambda >0\), with a constant \(C\) independent of \(\lambda \). Symbol \(\Psi \in \Delta _2^c\) will mean that the Legendre conjugate of \(\Psi \), that is, \(\Psi ^{*}(s):= \sup _{t>0}\{ st-\Psi (t)\}\), satisfies the \(\Delta _2\)-condition.
We will be using the following statement (see e.g. [9, Proposition 2]).
Proposition 2.1
Let M be a Young function and \((X,\mu )\) be the measurable space equipped with the measure \(\mu \). Then the expression
defines a complete norm on
for each \(\alpha \in (0,\infty )\). Moreover, all norms \(\Vert \cdot \Vert _{{L^\Psi (X,\mu ),\alpha }}\), \(\alpha \in (0,\infty )\) are equivalent.
2.2.2 Orlicz–Sobolev space
Let \(\Omega \subseteq \mathbb {R}^{n}\) be an open bounded domain, \(k\in \mathbf{N}\), and \(\Psi :[0,\infty )\rightarrow [0,\infty )\) be a nondecreasing convex continuous function such that \(\Psi (0)=0\) and \(\lim _{t\rightarrow \infty }\Psi (t)=+\infty \). The Orlicz–Sobolev space \(W^{k,p}(\Omega )\) is the linear set
equipped with the norm:
Here \(D^\alpha u\) means the distributional derivative of \(u\).
We define the space \(W^{k,\Phi }_\infty (\Omega )\), (respectively \(W^{k,\Phi }_L (\Omega )\)) as the completion of \(C^\infty (\bar{\Omega })\) (respectively \(Lip(\Omega )\)) in the norm of the space \(W^{k,\Phi } (\Omega )\).
2.2.3 Orlicz–Slobodetskii space \(Y^{\Psi ,\Phi }\)
Let \(\Omega \subseteq \mathbb {R}^{n}\) be an open bounded domain, \(\Psi ,\Phi :[0,\infty )\rightarrow [0,\infty )\) be nondecreasing convex continuous functions such that \(\Psi (0)=\Phi (0)=0\) and \(\lim _{t\rightarrow \infty }\Psi (t)=\lim _{t\rightarrow \infty }\Phi (t)=+\infty \). By \(Y^{\Psi ,\Phi }(\Omega )\) we denote the space of all \(u\in L^{\Psi }(\Omega )\), for which the seminorm
is finite. We equip it with the norm:
where
is the Luxemburg-type seminorm.
Analogously one can define \(Y^{\Psi ,\Phi }(u,M)\), \(I^{\Phi }(u,M)\), and \(J^{\Phi }(u,M)\), where \(M\subseteq {\mathbb {R}}^k\) is an arbitrary \(n\)-dimensional rectifiable set (\(n\le k\)) and instead of the Lebesgue measure we consider the \(n\)-dimensional Hausdorff measure \(\mathcal{H}^n\) defined on \(M\).
By \(Y^{\Psi ,\Phi }_\infty (\Omega )\) (respectively \(Y^{\Psi ,\Phi }_L (\Omega )\)) we will mean the completion of set
with respect to the norm \(\Vert u\Vert _{Y^{\Psi ,\Phi }(\Omega )}\). Analogously, if \(M\subseteq {\mathbb {R}}^k\) is an arbitrary \(n\)-dimensional rectifiable set (\(n\le k\)), by \(Y^{\Psi ,\Phi }_\infty (M)\) (resp. \(Y^{\Psi ,\Phi }_L(M)\)) we mean the completion of set
in the norm \(Y^{\Psi ,\Phi } (M)\), where by \(C^\infty ({M})\) we mean the set of functions which are defined on certain neighborhood of \(M\) in \({\mathbb R}^k\) and are smooth as functions on \({\mathbb {R}}^k\).
Remark 2.1
Let \(\Omega \subseteq \mathbb {R}^{n}\) be an open domain with locally Lipschitz boundary, \(\Psi (\lambda )=\Phi (\lambda )=|\lambda |^p\), \(1<p<\infty \). Then
which is the norm of \(u\) in the Slobodetskii space \(W^{1-\frac{1}{p},p}(\partial \Omega )\), see e.g. [33].
2.3 Basic assumptions
In the sequel we will be dealing with the following assumptions comming from papers by Kita [27, 28].
Definition 2.1
(Assumption A (Kita pair)) We assume that \(a, b: [0,\infty )\rightarrow [0,\infty )\) are strictly positive continuous functions such that
-
(a)
\(\int _0^1 a(s)/{s}\,ds <\infty \), \(\int _1^\infty \frac{a(s)}{s}\,ds =+\infty \);
-
(b)
\(b(\cdot )\) is non-decreasing, \(\lim _{s\rightarrow \infty }b(s)= +\infty \).
-
(c)
there exist constants \(c_1>0, s_0\ge 0\) such that
$$\begin{aligned} \int _0^s \frac{a(t)}{t}\,dt \le c_1 b(c_1s) \quad { \mathrm for}\, \mathrm{all }\quad s>s_0, \end{aligned}$$(2.4)and in the case \(s_0> 0\) mapping \(s\mapsto \frac{a(s)}{s}\) is bounded when \(s\ne 0\) is near to \(0\).
We define
Remark 2.2
We always have \( \Phi (t)\le \widetilde{C}\Psi (C_1t)\), with universal constants, so that \(\Phi \prec \Psi \). Moreover, the following three statements (a), (b), (c) are equivalent (see e.g. Proposition 5.1 in [25]):
-
(a)
\(\Psi \in \Delta _2^c\)
-
(b)
\(\Phi \) and \(\Psi \) are equivalent Orlicz functions,
-
(c)
\(\Phi \in \Delta _2^c\).
Example 2.1
Simple computation shows that pair \((\Phi (\lambda ),\Psi (\lambda ))= (\lambda (\mathrm{log} (2+\lambda ))^\alpha , \lambda (\mathrm{log} (2+\lambda ))^{\alpha +1})\) where \(\alpha > 0\) is a Kita pair.
2.4 Trace operator
Let us briefly recall basic claims from [23, Theorems 3.10 and 3.13]. The original formulation holds with \(u\in C^\infty (\bar{\Omega })\) however, the proof follows by the same arguments with no difference for \(u\in Lip ({\Omega })\) as well.
Theorem 2.1
([23], embedding theorem, inequalities in modulars) Let Assumption A be satisfied (see Definition 2.1) and let \(\Omega \) be a bounded domain with locally Lipschitz boundary. Then we have
-
(i)
If (2.4) holds with \(s_0=0\), then for every \(u\in Lip ({\Omega })\),
$$\begin{aligned}&\int _{\partial \Omega } \Psi ( |u(x)|)\mathcal{H}^{n-1}(dx) + I^{\Phi }(u,\partial \Omega )\nonumber \\&\quad \le C\left( \int _\Omega \Psi (C_1|u (x)|)\, dx + \int _\Omega \Psi (C_2|\nabla u (x)|)\, dx\right) , \end{aligned}$$(2.6)with constants \(C\), \(C_1, C_2\) independent of \(u\).
-
(ii)
If (2.4) holds with some \(s_0>0\), then for every \(u\in Lip ({\Omega })\),
$$\begin{aligned}&\int _{\partial \Omega } \Psi (|u(x)|)\mathcal{H}^{n-1}(dx) + I^{\Phi }(u,\partial \Omega )\nonumber \\&\quad \le C\left( \int _\Omega \Psi (C_1|u (x)|)\, dx + \int _\Omega \Psi (C_2|\nabla u (x)|)\, dx + \int _\Omega |\nabla u (x)|\, dx\right) ,\nonumber \\ \end{aligned}$$(2.7)with constants \(C,C_1,C_2\) independent of \(u\).
Theorem 2.2
([23], embedding theorem, inequalities in norms) Let Assumption A be satisfied (see Definition 2.1) and let \(\Omega \) be a bounded domain with a locally Lipschitz boundary. Then there exists a constant \(D\) such that for every \(u\in Lip ({\Omega })\) we have
2.4.1 Trace operator
Let the assumptions in Theorems 2.1 and 2.2 be satisfied and let \(u\in W^{1,\Psi }_{Lip} (\Omega )\). Consider any sequence \(u_m\in L (\bar{\Omega })\) convergent to \(u\) in the norm of the space \(W^{1,\Psi }(\Omega )\). Then \(\{ u_m\}\) is a Cauchy sequence in \(Y^{\Psi ,\Phi }(\partial \Omega )\) (convergence in the norm), so that it converges some element \(\bar{u}\in Y^{\Psi ,\Phi }_L(\partial \Omega )\). It is easy to observe that \(\bar{u}\) is independent of the choice of the sequence \(\{ u_m\}\subseteq L (\bar{\Omega })\), converging to \(u\). It allows to extend the standard definition of the trace operator:
where the convergence holds in the norm of \(Y^{\Psi ,\Phi }(\partial \Omega )\).
As a consequence we obtain the following result.
Theorem 2.3
([23], embedding theorem) Let Assumption A be satisfied (see Definition 2.1) and let \(\Omega \) be a bounded domain with a locally Lipschitz boundary. Then trace operator \({{\mathrm{Tr}}}: W^{1,\Psi }_L (\Omega )\mapsto Y^{\Psi ,\Phi }_L(\partial \Omega )\) is well defined by expression (2.9) and there exists a constant \(D\) such that for every \(u\in W^{1,\Psi }_L (\Omega )\) we have
3 Heat kernel estimates
Let \(u\in Lip (\partial \Omega )\). We will define function \(\widetilde{u}\in Lip (\Omega )\) such that \({{\mathrm{Tr}}}\widetilde{u}=u\) using the Gaussian kernel. Let
be the heat kernel. Then \(E\) obeys the following properties
-
1.
$$\begin{aligned} \int _{{\mathbb R}^{n-1}} E(x,t)\,dx =1 \quad \mathrm{for \, every }\ t>0, \end{aligned}$$
-
2.
$$\begin{aligned} \lim _{t\rightarrow 0} \int _{{\mathbb R}^{n-1}} \phi (x)E(x,t)\,dx =\phi (0) \ \quad \hbox {for every}\ {\phi \in Lip_0({\mathbb R}^{n-1})}, \end{aligned}$$
-
3.
$$\begin{aligned} E_t(x,t)=\Delta _xE(x,t). \end{aligned}$$
We will start our construction with the case when \(\Omega =Q\), and assuming that \(u\in Lip_0 (Q')\).
We define
where \(g*u\) is the usual convolution. We have the following observation.
Lemma 3.1
For any \(u\in Lip_0 (Q')\) we have \(\widetilde{u}\in Lip (\bar{Q})\). In particular \({{\mathrm{Tr}}}\widetilde{u}=u\).
The remaining estimates and our final result will be established in several steps presented in in the sequel.
4 Estimates of function \(\widetilde{u}\)
4.1 Presentation of main results
We start with the following result.
Proposition 4.1
([23]) Let \(u\in Lip (\mathbb {R}^{n-1})\) be supported in \(Q^{'}\), and \(\widetilde{u}\) be defined by (3.1). Then for any \(\lambda >0\) and any convex function \(R: [0,\infty )\rightarrow [0,\infty )\) and for any \(T>0\), we have
In particular, if \(R\) is a N-function, we have \(\Vert \widetilde{u}\Vert _{L^{R}(Q)}\le \Vert u\Vert _{L^{R}(Q')}.\)
Our goal is to show that in some cases the function \(\widetilde{u}\) can have better integrability properties than \(u\). As main result of this section we obtain the following lemma.
Lemma 4.1
Let \(\widetilde{u}\) be given by (3.1), where \(u\in Lip_0 (Q')\). Moreover, let \((\Phi ,\Psi )\) be as in Assumption A (see Definition 2.1) and suppose that \(\Phi \) satisfies the following estimate:
and continuous, nondecreasing function \(G\) such that \(\int _0^1 G(c|\ln t|)\,dt <\infty \) for any constant \(c>0\). Then
where constant \({C}>0\) above is independent of \(u\).
Remark 4.1
-
1.
The condition (4.2) implies that \(\Phi \) satisfies \(\Delta _2\)-condition near infinity, i.e. there exists \(N>0\) and \(C>0\) such that \(\Phi (2y)\le C\Phi (y)\) whenever \(y>N\).
-
2.
Assume that \(\Phi (s)= s^{\alpha }h(|\ln s|)\) where \(h\) is continuous, positive for positive arguments, nondecreasing and satisfies \(\Delta _2\)-condition near \(\infty \) (in the sense of definition given above). Then
$$\begin{aligned} \Phi (xy)\precsim 1+ G(x)+ G(x)\Phi (y), \mathrm{where}\,\, G(x)=\mathrm{max}\{ x^{\alpha }, \Phi (x))\}. \end{aligned}$$Indeed, this follows from the following estimates:
$$\begin{aligned} \Phi (xy)&=x^{\alpha }y^{\alpha }h( |\ln x|+|\ln y|)\le x^{\alpha }y^{\alpha }\left( h( 2|\ln x|) +h(2|\ln y|) \right) \\&\precsim x^{\alpha }y^{\alpha }( 1+ h(|\ln x|) +h(|\ln y|)) \\&= y^{\alpha }\Phi (x) + x^{\alpha }\Phi (y) + x^{\alpha }y^{\alpha } = y^{\alpha }(\Phi (x) + x^{\alpha }) + x^{\alpha } \Phi (y)\\&\precsim G(x)y^{\alpha } + G(x) \Phi (y) \\&\precsim G(x)+G(x)\Phi (y). \end{aligned}$$ -
3.
In the original formulations of Lemmas 4.1 and 4.1 we assume \(u\in C^\infty \) instead of \(u\) being Lipschitz. The statements we formulate now hold true without any changes in their proofs.
4.2 Proof of Lemma 4.1
The proof will be proceeded by the known results of Kita [28, Theorems 2.1 and 2.7] and sequence of lemmas where we derive certain pointwise estimates.
Theorem 4.1
([28]) Let \(a(s),b(s),\Phi (t),\Psi (t)\) be as in Assumption A (see Definition 2.1), with \(s_0\ge 0\). Then there exist constants \(C_1, C_2, C_3>0\) such that
for all \(f\in L^1 (\mathbb R^{n})\), where \(Mf(x)\) is the Hardy–Littlewood maximal function of \(f(x)\).
Lemma 4.2
Let \(\beta >-2\) and
Then there exist constants \(C_0,C_1\) (depending on \(\beta \)) such that for every \(y\in {\mathbb R}_+\) we have
Proof
Function \(F_\beta \) is increasing up to certain \(x_\beta \) and it is bounded. Therefore inequality follows with \(C_0=\sup F< \int _0^\infty q^{\beta +1}e^{-q}\,dq \), \(C_1=x_\beta \). \(\square \)
Lemma 4.3
Let \(s\in {\mathbb R}_+\), \(t>0, \alpha >0\) and
Then for any \(s,t_0,r>0\) we have
where \(F_\beta (x)\) is given by (4.5), \(C_1\) is the same as in Lemma 4.2.
Proof
We start with the case \( 0<r<\frac{t_0}{2}\). Let us find the negative integer \(k\le -2\) such that \( t_02^k\le r\le t_02^{k+1}\), so that for \(B=B_\alpha (s,t_0,r)\) we have
because when \(t\in (t_0-r, t_0+r)\) we have \(\frac{t_0}{2}\le t \le \frac{3t_0}{2}\) and \(|(t_0-r,t_0+r)|=2r\le 2^{k+2}t_0\).
In case \(\frac{t_0}{2}\le r<t_0\), we have (we change variables, substituting \(q=\frac{s^2}{4t}\))
Finally, if \( r\ge t_0\), we proceed similarly as above, to get
then we use Lemma 4.2 (note that \(\frac{s^2}{8r}\le \frac{s^2}{8t_0}=:y\)) to get in this case
Therefore the estimate holds in all considered cases. \(\square \)
Lemma 4.4
Let \(u\in L^1(Q')\), \(Q'=[0,1]^{n-1}\), and \(\widetilde{u}\) be given by (3.1), \(M^2\) is as in (2.1). Then there exists constant \(C_1>0\) such that for any \( t_0>0\) we have
where
The estimate holds with some constant which is independent of \(u, x, t\).
Proof
Applying Fubini’s theorem we obtain
Now it suffices to apply Lemma 4.3. \(\square \)
Remark 4.2
We observe that under the notation in Lemma 4.4 we have
-
1.
when \(u\) is nonnegative then \((\mathcal{T}_0 u)(x,t) =\widetilde{u}(x,t)\), in general \((\mathcal{T}_0 u)(x,t) =\widetilde{|u|}(x,t)\) (see 3.1);
-
2.
\(C_1\) is as in Lemma 4.2 with \(\beta = \frac{n-5}{2}\).
Lemma 4.5
Let \(u\in L^1({\mathbb R}^{n-1})\) and let \((\mathcal{T}_1 u)(x,t_0)\) be as in Lemma 4.4. Then for any convex function \(R\) and every \(t_0>0\) we have
where \(\widetilde{C}=\int _{{\mathbb R}^{n-1}} |z|^{-(n-1)}F_{\frac{n-5}{2}}(|z|^2)dz\).
Proof
We have
and \(\widetilde{C}= \int _{{\mathbb R}^{n-1}}\frac{1}{|x-y|^{n-1}} F_{\frac{n-5}{2}}( \frac{|x-y|^2}{4t_0} )\,dy=\int _{{\mathbb R}^{n-1}} |z|^{-(n-1)}F_{\frac{n-5}{2}}(|z|^2)dz\). Therefore by Jensen’s inequality
Integrating the above expression over \({\mathbb R}^{n-1}\), then applying Fubini’s theorem, we get
This finishes the proof of the lemma. \(\square \)
Lemma 4.6
Let \(\mathcal{T}_2\) be as in Lemma 4.4, \(R\) be an arbitrary convex function and \(Q'=[0,1]^{n-1}\). Then for any \(t>0\) and measurable function \(u\) supported in \(Q^{'}\) we have
with constant \(C>0\) independent of \(u\).
Proof
If \(x\in Q'\) and \(y\in Q'\), then \(|x-y|\le \sqrt{n-1}\). We have
It suffices to show that
where \(C_0\) is independent in \(u\). We start with the case \(t< \frac{1}{64}\). For this, let
We have
Let us fix \(x^{'}=(x_2,\ldots ,x_{n-1})\in Q^{'}\) and consider the mapping
We will show that \(f_1(x_1)\) takes its minimal value at \(x_1=1/2\). Indeed, let \(x=(x_1,x^{'})\in (0,1)\times (0,1)^{n-2}\). We have
Define for \(s\in (0,1)\):
Note that as for \(s_1\le s_2\) we have \(\mathcal{C}(s_1)\subseteq \mathcal{C}(s_2)\), therefore function \(s\mapsto h(s)\) is nondecreasing. Moreover, let
Then \(|A(x_1)|+ |B(x_1)|=1\), sets \(A(x_1)\), \(B(x_1)\) are disjoint and \(\mathrm{inf}\{ h(|z_1|^2): z_1\in B(x_1)\} \ge h(\frac{1}{4})\ge \mathrm{sup}\{ h(|z_1|^2): z_1\in [-\frac{1}{2},\frac{1}{2}]\}\). As we have
sets \(B(x_1)\) and \([-\frac{1}{2},\frac{1}{2}]{\setminus } A(x_1)\) have the same measure, therefore the expression in brackets \(\{ \dots \}\) is nonnegative. Therefore \(f_1(x_1)\) achieves its minimal value when \(\{ \dots \}=0\), in particular when \(A(x_1)=[-\frac{1}{2},\frac{1}{2}]\), \(B(x_1)=\emptyset \), i. e. in case when \(x_1=\frac{1}{2}\).
By an obvious modification of the above argument we have
and right hand side is nonzero provided that \(2\sqrt{t}<|(1/2,\ldots ,1/2)|=\sqrt{n-1}/2\). When \(t<1/16\) this is always satisfied and in that case \(C(x,t)>0\). Moreover, by simple observation we have
where \(\omega _{n-2}\) is measure of unit sphere in \({\mathbb R}^{n-2}\) is the case \(n>2\) and \(\omega _0=2\). On the other hand
This easily implies
By Jensen’s inequality, Fubini’s theorem and the above estimates we can estimate further:
where \(C\) is independent on \(u\).
When \(t>\frac{1}{64}\) the computations become simpler as then we have
which implies
\(\square \)
We are now in position to prove Lemma 4.1.
Proof of Lemma 4.1
Let \(\lambda >0\) and denote
We apply Theorem 4.1 to the internal integral, then Fubini Theorem, to get
This is further estimated with the help of Lemma 4.4:
Applying Proposition 4.1 and Lemma 4.5 with \(R=\Phi \), we obtain
Moreover, according to Lemma 4.6, we have
with some general constants \(C_1,C_2,C_3>0\). This together with (4.2) gives \(\mathcal{C}\lesssim 1+ \int _{Q'}\Phi ( \frac{ |u(x)|}{\lambda })\,dx\). To get (4.3) it suffices to note that if \(s_0>0\), then
The last estimate follows from Proposition 4.1. This implies (4.3). Inequality (4.4) follows directly from (4.3) and Proposition 2.1 after the substitution \(\lambda = \Vert u\Vert _{L^\Phi (Q')}\). \(\square \)
5 Estimates of function \(\nabla _x\widetilde{u}\)
5.1 Presentation of main results
Our approach will be based on the following results.
Proposition 5.1
(Lemma 3.3, [23]) For any \(k\in \{ 1,\ldots , n-1\}\), any \(0<\varepsilon <\frac{1}{2}\), and any \(u\in Lip_0 ( Q')\) such that \({{\mathrm{dist}}}({{\mathrm{supp}}}\,u,\partial Q')\ge \varepsilon \), we have
Proposition 5.2
([23]) For any convex function \(R\), any \(0<\varepsilon <\frac{1}{2}\) and any function \(u\in Lip_0 ( Q')\) such that \(\mathrm{dist}({{\mathrm{supp}}}u,\partial Q')\ge \varepsilon \) we have
with \(C_1=2\sqrt{n-1}\int _{{\mathbb R}^{n-1}}|w|^2e^{-|w|^2}\,dw\), \({C}_2=\frac{2\sqrt{n-1}}{\sqrt{2\pi e}\cdot \varepsilon }\).
In particular, when \(\nabla _xu\) denotes the spatial gradient of \(u\), we have
Proof
We use argument preceding (3.5) in the proof of Lemma 3.2 in [23], to get the slightly more precise statement. \(\square \)
Corollary 5.1
([23]) If \(u\in Y^{\Phi ,\Phi }_{L}(Q')\) is supported in the interior of \(Q'\) then the spatial derivative of \(\widetilde{u}\) satisfies \(|\nabla _x\widetilde{u}|\in L^\Phi (Q)\).
Remark 5.1
As before we remark that the original formulations of Lemmas 5.1 and 5.2 and Corollary 5.1 deal with \(u\in C_0^\infty (Q')\) instead of \(u\in Lip_0(Q')\) but their statements hold with Lipschitz \(u\) by the same arguments.
Remark 5.2
Estimate (5.1) shows that
where \(\omega (x,y)=\frac{1}{|x-y|}\). This implies the estimates in the norms
where \(Y^{R,R}_\omega (Q^{'})\) is weighted Orlicz–Slobodetskii space introduced in [11, 12]. In particular more precise statement than that in Corollary 5.1 holds.
Corollary 5.2
Let \(Y_{\omega ,L}^{\Phi ,\Phi }(Q')\) denote the completion of Lipschitz functions in \(Y_{\omega }^{\Phi ,\Phi }(Q')\). If \(u\in Y_{\omega ,L}^{\Phi ,\Phi }(Q')\) is supported in the interior of \(Q'\) then the spatial derivative of \(\widetilde{u}\) satisfies \(|\nabla _x\widetilde{u}|\in L^\Phi (Q)\).
Note that we have \(Y^{\Phi ,\Phi }(Q')\subseteq Y_\omega ^{\Phi ,\Phi }(Q')\).
Our main goal is to show that in general stronger property follows. Namely, that under an assumption \(u\in Y^{\Psi ,\Phi }_L(\partial \Omega )\), where \(u\) is supported in the interior of \(Q'\) and \((\Psi ,\Phi )\) is as in Assumption A (see Definition 2.1), we have \(|\nabla _x\widetilde{u}|\in L^\Psi (Q)\). Let us recall that in general we have \(\Phi \prec \Psi \) (see Remark 2.2) and it may happen that \(\Phi \not \sim \Psi \).
The main result of this subsection reads as follows.
Lemma 5.1
Let \(0<\varepsilon <1/2\) and \(\widetilde{u}\) be given by (3.1), where \(u\in Lip_0 (Q')\) and \(\mathrm{dist}({{\mathrm{supp}}}u,\partial Q')\ge \varepsilon \). Moreover, let \((\Phi ,\Psi )\) be as in Assumption A (see Definition 2.1) and \(\Phi \) satisfies the following estimation:
where \(G\) is continuous, nondecreasing, locally bounded and such that
for any \(c>0\). Then for every \(\lambda >0\)
Remark 5.3
Constant bounds in the above estimations depend on \(\epsilon >0\).
5.2 Proof of Lemma 5.1
First we prove several auxiliary claims. We shall proceed quickly when the proofs are similar to that of the previous subsection.
Lemma 5.2
Let \(s\in {\mathbb R}_+\), \(t>0, \alpha >0, \gamma \ge 0\) and
Then for any \(s,t_0,r>0\) we have
where \(F_\beta (x)\) is given by (4.5), \(C_2=C_2(\alpha ,\gamma )>0\) is a given constant.
Proof
The proof is the little modification of Lemma 4.3 with the difference that instead of \(\widetilde{E}_{\alpha }(s,t)\) we now deal with \(\widetilde{E}_{\alpha ,\gamma }(s,t)=\widetilde{E}_{\alpha }(s,t)( \frac{s^2}{4t})^\gamma \). Repeating the proof of Lemma 4.3 we observe that
in case \(0< r< \frac{t_0}{2}\). In case \(\frac{t_0}{2}\le r<t_0\) we have the estimation
while in case \(r\ge t_0\) we compute that
Using Lemma 4.2 dealing with \(y=\frac{s^2}{8t_0}\) and \(\beta =\alpha +\gamma -2>-2\), we obtain that when \(\frac{t_0}{2}\le r\)
with some positive constant \(C_2>0\) depending on \(\alpha +\gamma \). This completes the proof.
\(\square \)
Lemma 5.3
Let \(u\in Lip_0 (Q')\) and \(\widetilde{u}\) be given by (3.1). Then for any \( t_0>0\) we have
where constant \(C_2>0\) does not depend on \(u\).
Proof
According to Proposition 5.1 we have
Now it suffices to estimate \( \frac{1}{2r}\int _{(t_0-r,t_0+r)} P(|x-y|,t)\chi _{t>0}\,dt\) with help of Lemma 5.2 with \(\alpha =\frac{n-1}{2}\), \(\gamma =1\). \(\square \)
Lemma 5.4
Let \(i\in \{ 1,2,3\}\) and \((\mathcal{P}_iu)(x,t)\) be defined by:
where \(C_2>0\) is the same as in Lemma 5.3 (it does not depend on \(u\)). Let \(0<\epsilon <1/2\) be given and \(u\in Lip_0(Q')\) be such that \(\mathrm{dist}( \mathrm{supp u},\partial Q^{'})\ge \epsilon \). Then we have for any \(t_0>0\)
Moreover, let \(R\) be such that
and \(G\) is nonincreasing and locally bounded. Then for any convex function \(R\) we have for any \(T>0\)
where \(i\in \{ 0,1,2\}\), with some constant \(C>0\) which is independent of \(u\).
Proof
In this proof constant \(C>0\) will denote some general constant independent of \(u\). It can be different even in the same line. We start with the estimate of \(\mathcal{P}_0\). For this, we apply arguments from the proof of Lemma 3.2 in [23], to get
whenever \(T>0\). For reader’s convenience we submit them. As \(C_1:= \int _{\mathbb {R}^{n-1}} P(|x-y|,t)dy>0\) does not depend on \(t\), we have from Jensen’s Inequality:
Simple computation shows that for any \(s>0\)
and we apply the estimate (5.5 5.6) on \(R\) involving \(\bar{x}=C_1,\bar{y}=\frac{|u(y)-u(x)|}{|x-y|}\).
The proof of the estimate of \(\mathcal{P}_1\) goes more or less along the lines of that of Lemma 4.5. Namely, we observe that for \(x\in Q'\)
where we choose \(\widetilde{C}=\int _{{\mathbb R}^{n-1}}\frac{1}{|x-y|^{n-1}}F_{\frac{n-3}{2}} ( \frac{|x-y|^2}{4t})\,dy =\int _{{\mathbb R}^{n-1}}\frac{1}{|z|^{n-1}}F_{\frac{n-3}{2}} ( {|z|^2})dz\). Therefore
Now it suffices to verify that for small \(s\),
Now let us estimate \(\mathcal{P}_2\). We have (with \(C=\frac{1}{4C_2}\)),
Where we chose at first the positive constant
We can assume that \(C\) in the condition \(t<C|x-y|^2\) is bigger than one and that \(C(x,t)\) is positive. This is because if \(C_1<C_2\) we have \(\mathcal{A}(C_1)\le \mathcal{A}(C_2)\), therefore if we enlarge \(C\), then we get \(R( (\mathcal{P}_3u))\le \mathcal{A}(C)\), which is sufficient for our analysis.
Observe that
when \(t\) is sufficiently small. On the other hand we always have \(C(x,t)\lesssim (1+|\ln t|),\ \hbox {for every}\ t\in (0,1)\), therefore
where \(C_0\) does is some general positive constant. Combining this with Jensen’s inequality we obtain the following inequality for small \(t\):
Let \(\bar{x}:= C(1+|\ln t|), \bar{y}:= \frac{|u(x)-u(y)|}{|x-y|}\). Using the condition (5.5 5.6) on \(R(\bar{x}\bar{y})\) and estimating further, we get
When \(T\ge t\ge C_0\), the estimates become simples as then we have
where we chose \(b(x)=\int _{Q'} \sqrt{C/C_0} \frac{1}{|x-y|^{n-2}}\,dy\sim 1\) (as \(x\in Q^{'}\)). Therefore and by Jensen’s inequality
After integrating it over \((C_0,T)\times Q^{'}\) and using (5.5 5.6) again, we obtain
which finishes the proof. \(\square \)
Proof of Lemma 5.1
We note that for an arbitrary \(\lambda >0\) we have by Theorem 4.1,
Moreover, \(|f(x,t)|\le M^2f(x,t)\) and \(\int _{Q} |h(x,t)|\,dx dt \le \Phi ^* (1) + \int _{Q}\Phi (|h(x,t)|)\,dx dt\), hence \(\mathcal{L}_1\prec 1+ \mathcal{L}_2\) and it remains to estimate the expression \(\mathcal{L}_2\).
According to Lemma 5.3 we have (with the same notation)
Applying Lemma 5.4 with \(R=\Phi \) and observing that \(\Phi (\sum _{j=1}^4 a_j)\le \frac{1}{4}\sum _{j=1}^4\Phi (4a_j)\lesssim 1+ \sum _{j=1}^4\Phi (a_j)\), we arrive at
This gives (5.3). The choice of \(\lambda _0 = I^\Phi (u,Q') +\Vert u\Vert _{L^\Phi (Q')}\) implies
which together with Proposition 2.1 gives (5.4). This ends the proof of the lemma. \(\square \)
6 Estimates of function \(\partial _t\widetilde{u}\)
6.1 Presentation of results and discussion
Our goal here is to continue our estimates of heat extension operator, dealing now with the time derivative of \(\widetilde{u}\).
We start with the following lemma. The Lemma was originally proven for \(u\in C_0^\infty (Q^{'})\). We remark that assumption \(u\in Lip_0(Q')\) does not change final conclusion and the proof under such assumption follows by the same arguments.
Lemma 6.1
([23]) Let \(u\in Lip_0(Q')\), \(\widetilde{u}\) be given by (3.1), \(0<\varepsilon <\frac{1}{2}\) and let \({{\mathrm{dist}}}({{\mathrm{supp}}}u,\partial Q')\ge \varepsilon \). Then we have
with \(\widetilde{C}_1= 2n\), \(\widetilde{C}_2=\frac{1}{\sqrt{\pi }}(\sqrt{3})^3e^{-\frac{9}{4}}\cdot (n-1)\).
A wishful thinking would expect the inequality of type
dealing with an arbitrary convex function \(R\). It seems quite difficult to prove such an inequality. However, we have obtained such result under the special assumption stated below.
Definition 6.1
(Assumption B)
-
1.
\(R(\lambda )=\lambda P(\lambda )\), where \(P(ab)\prec 1+ P(a) + P(b),\)
-
2.
\(P\) is nondecreasing,
-
3.
Function \(\frac{P(\lambda )}{\lambda }\) is nonincreasing for large arguments.
Remark 6.1
([23])
-
1.
Condition 1., 2., and 3. above imply \(P(x^n)\le C_nP(x)\) for big arguments. It gives inequality \(P(y)\le C_\kappa y^\kappa \) for large arguments, with any \(\kappa >0\).
-
2.
Let us consider \(P(\lambda )= (\ln (2+\lambda ))^\alpha \), with any \(\alpha >0\). Then \(P\) satisfies 1, 2, and 3. Moreover, \(P(\lambda )\prec 1+ \lambda ^\beta \), with any \(\beta >0\).
-
3.
We have that \(\int _0^{a}P(\frac{1}{s})\,ds <\infty \) for every \(a>0\).
We obtained the following result (formulated originally for \(u\in C_0^\infty ( Q')\)).
Proposition 6.1
([23]) If \(R\) satisfies an Assumption B (see Definition 6.1), then for any function \(u\in Lip_0 ( Q')\) we have
Remark 6.2
It follows from the proof of Proposition 6.1 presented in [23] that when
then for any \(u\) supported in \(Q'\) and any \(R\) satisfying Condition B, we have
for every \(0<T<\infty \).
It is clear from Proposition 6.1 that under certain assumptions on function \(R\), condition \(u\in Y_L^{R,R}(Q')\) implies \(\partial _t\widetilde{u}\in L^{R}(Q)\). Following our previous schema, we would like to prove that the condition \(u\in Y^{\Phi ,\Phi }(Q')\) implies \(\partial _t\widetilde{u}\in L^{\Psi }(Q)\), where \((\Phi ,\Psi )\) is as in Assumption A. We do not know if this is true in general. However, we have the following result.
Lemma 6.2
Let \(0<\epsilon <1/2\), \(u\in Lip_0 ( Q')\) is such that \(\mathrm{dist} ( \mathrm{supp} u,\partial Q^{'})\ge \epsilon \) and \(\widetilde{u}\) is given by (3.1). Moreover, let \((\Phi ,\Psi )\) be as in Assumption A (see Definition 2.1) and \(\Phi \) satisfies Assumption B (see Definition 6.1). Then we have for every \(\lambda >0\):
Remark 6.3
Note that under the assumptions of Lemma 6.2 we have
and parameter \(\delta >0\) above can be taken arbitrarily. Moreover, the above inequality is very close to the inequality:
We do not know if (6.4) holds with \(\delta =0\).
We are now in position to prove the presented result.
6.2 Proof of Lemma 6.2
We start with the following result which extends Lemma 5.2.
Lemma 6.3
Let \(s\in {\mathbb R}_+\), \(t>0, \alpha >0,\) \(v(x)=x+x^3\) and
Then for any \(s,t_0,r>0\) we have
with some constant \(C_3>0\).
Proof
We note that under an assumption of Lemma 5.2 we have \(\widetilde{E}_{\alpha ,w}(s,t) = \widetilde{E}_{\alpha ,\frac{1}{2}}(s,t)+ \widetilde{E}_{\alpha ,\frac{3}{2}}(s,t)\). The estimate follows from Lemma 5.2 by summing up the estimates for \(\widetilde{E}_{\alpha ,\frac{1}{2}}(s,t)\) and \(\widetilde{E}_{\alpha ,\frac{3}{2}}(s,t)\), after we note that \(F_\beta (x)\lesssim 1\) for large arguments, so that \(F_\beta (\frac{s^2}{4t_0})\) can be estimated by constant when \( t_0<\frac{s^2}{4C_3}\). On the other hand \(F_\beta (x)\sim x\) for small \(x\). \(\square \)
Lemma 6.4
Let \(u\in Lip_0 (Q')\) and \(\widetilde{u}\) be as in Lemma 6.2 and let \(S(s,t)\) be as in Lemma 6.1. Then for any \( t>0\),
with some constant \(C_3,\) which is not dependent on \(u\).
Proof
According to Lemma 6.1, we have
Using the notation of Lemmas 6.1 and 6.3 we have \(S(s,t)=\widetilde{E}_{\frac{n}{2},v}(s,t)\). Therefore the result follows from Lemma 6.3 applied with \(\alpha =\frac{n}{2}\). \(\square \)
We are now to establish our crucial estimates for \(\partial _t\widetilde{u}\). We have the following result.
Lemma 6.5
Let \(u\in Lip_0 (Q')\) and \(\widetilde{u}\) be as in Lemma 6.2, \(i\in \{ 1,2,3\}\) and \((\mathcal{C}_iu)(x,t)\) be the same as in Lemma 6.4 and \(R\) satisfies Assumption B (see Definition 6.1). Then we have
Proof
Denote for simplicity \(h(x,y):=\frac{|u(x)-u(y)|}{|x-y|}\). The proof will be divided into several steps.
Step 1 (proof of 6.5). The estimate for \(i=1\) is a consequence of Remark 6.2 as \((\mathcal{C}_1 u)(x,t) = Bu (x,\frac{3}{2}t)\).
Step 2 (reduction argument). We note that for \(\theta := 2\alpha -1\), \(\alpha \in (1/2,1)\):
By the convexity argument \(R(\sum _{i=1}^3\mathcal{B}_iu)\le \frac{1}{3}\sum _{i=1}^3R(3\mathcal{B}_iu)\) and \(R(Ca)\lesssim 1+ R(a)\), therefore it suffices to prove (6.5) with \(\mathcal{B}_iu\) instead of \(\mathcal{C}_2u\).
Step 3 (proof of (6.6) and (6.7) with \(\mathcal{B}_1u\) instead of \(\mathcal{C}_2u\)).
For this purpose we note that
Hence \(R(\mathcal{B}_1u)\lesssim 1+ R(t^{-\alpha })\), consequently
Step 4 (proof of (6.6) and (6.7) with \(\mathcal{B}_2u\) instead of \(\mathcal{C}_2u\)).
As \( \int _{Q'\cap \{ y: t\ge \frac{|x-y|^2 }{4C_3}\}} \frac{1}{|x-y|^{n-2}} t^{-1/2} \,dy \sim 1 \), we can apply Jensen’s inequality to get
with some constant \(c>0\). We have from Assumption B \(P (\frac{ch}{\sqrt{t}})\lesssim 1+ P(h) + P(\frac{1}{\sqrt{t}})\lesssim P(\frac{1}{\sqrt{t}})\) on the considered set of integration. Therefore \(R (\frac{ch}{\sqrt{t}}) \lesssim \frac{1}{\sqrt{t}} h P (\frac{ch}{\sqrt{t}})\lesssim \frac{1}{\sqrt{t}} h P (\frac{1}{\sqrt{t}})\) on this set. This implies
where \(A:= \{ Q'\times Q'\times (0,1): t\ge \frac{|x-y|^2 }{4C_3}, ( \frac{1}{\sqrt{t}})^\theta < h< \frac{1}{\sqrt{t}}\}\). Now we estimate the internal integral in brackets \(\{\dots \}\) denoting it by \(X\). Note that when \(\frac{1}{\sqrt{t}}\in (h, h^{1/\theta })\), we get \(P(h)\lesssim P(\frac{1}{\sqrt{t}})\lesssim 1+ P(h)\) (see Remark 6.1, part 1). This (and the conditions \(h<\frac{1}{\sqrt{t}}\), \(h>1\)) imply that \(X\lesssim P(h)\int _{(\frac{1}{h})^{\frac{2}{\theta }}}^{\frac{1}{h^2}} \frac{1}{t}\,dt \sim P(h)|\ln h|\). On set of integration we have the condition \(h\le \frac{C}{|x-y|}\) with some constant \(C>0\). Therefore both inequalities hold:
which implies our assertion.
Step 5 (proof of (6.6) and (6.7) with \(\mathcal{B}_3u\) instead of \(\mathcal{C}_2u\)).
For this purpose we apply inequality
which is obtained by similar arguments as the ones to get (6.9). Observing that \(P(\frac{ch}{\sqrt{t}})\lesssim P(h)\), we get, by similar computations as in previous step that
where \(B:= \{ Q'\times Q'\times (0,1): t\ge \frac{|x-y|^2 }{4C_3}, h\ge \frac{1}{\sqrt{t}}\}\). This easily gives the desired estimates.
Step 6 (proof of (6.8)).
To this goal we again use the fact that when \(t\) is sufficiently small, i.e. \(t<t_0\) for some \(t_0\in (0,1)\), we have
\(\int _{Q'\cap \{ y: t<\frac{|x-y|^2 }{4C_3}\}} \frac{\sqrt{t}}{|x-y|^{n}} \,dy \sim 1\) and then we apply Jensen’s inequality to get
recalling that \(\theta = 2\alpha -1\). Now we estimate \(A(x,t)\) and \(B(x,t)\) separately. To deal with \(A\), we note that (as \(h\le (\frac{1}{\sqrt{t}})^\theta \)), we have \(P( \frac{h}{\sqrt{t}} )\lesssim 1+ P(h) + P(\frac{1}{\sqrt{t}})\lesssim P(\frac{1}{\sqrt{t}})\). Consequently
Now we estimate the integral in brackets \(\{\cdot \}\) denoted by \(Y\). Note that on set of integration we have \(\frac{1}{|x-y|}\lesssim \frac{1}{\sqrt{t}}\), moreover, \(P(\lambda )\lesssim \lambda ^\varepsilon \) for arbitrary \(\varepsilon >0\) (see Remark 6.1). Taking this into account we get:
where \(\beta = 2(1-\alpha -\varepsilon )>0\), (it is enough to take sufficiently small \(\varepsilon \)). This implies \(I_1\lesssim 1\).
To estimate the term with \(B\) we note that on set of integration we have \(P( \frac{h}{\sqrt{t}})\lesssim P(h)\), so that
The estimates when \(t>t_0\) become simpler as on set if integration we have \(\sqrt{{4C_3} t_0}<|x-y| \), therefore we omit them. Lemma is proved. \(\square \)
Proof of Lemma 6.2
The proof if obvious modification of the proof of Lemma 5.1 and is based on Lemmas 6.4 and 6.5. \(\square \)
7 Final results
We are now to present our main results. For this purpose we introduce the new space of functions. Let \(\Omega \subseteq {\mathbb R}^n\) be a Lipschitz boundary domain. By \(Y^{\Psi ,\Phi }_{log}(\partial \Omega )\) we will mean the modification of the space \(Y^{\Psi ,\Phi }(\partial \Omega )\), where the seminorm:
is substituted by
where \(\sigma \) is the \(n-1\)-dimensional Hausdorff measure on \(\partial \Omega \). By \(Y^{\Psi ,\Phi }_{L,log}(\partial \Omega )\) we will mean the completion of set \(\{ u\in Lip (\partial {\Omega })\cap Y^{\Psi ,\Phi }_{log}(\partial \Omega ) \}\) in the norm of \(Y^{\Psi ,\Phi }_{log}(\partial \Omega )\).
Our first final result reads as follows.
Theorem 7.1
(Theorem about extension) Let \(\Omega \subseteq {\mathbb R}^n\) be a bounded Lipschitz boundary domain, \(u\in Y^{\Phi ,\Phi }_{L, \log }(\partial \Omega )\), where \((\Psi ,\Phi )\) is as in Assumption A (see Definition 2.1) and let \(\Phi \) satisfy Assumption B (see Definition 6.1). Then there exists function \(\widetilde{u}\in W^{1,\Psi }_L(\Omega )\) such that \({{\mathrm{Tr}}}\widetilde{u}=u\), moreover, we have
Consequently
with constant \(C\) independent of \(u\).
Remark 7.1
As a consequence of Theorem 7.1 we obtain inequality
with an arbitrary \(\delta >0\) independent on \(u\). Note that \(\int _{\partial \Omega } \Phi (|{u}|)d\sigma (x)\prec 1+ \int _{\partial \Omega } \Psi (|{u}|)d\sigma (x)\), so this inequality is very close to the following one:
and the above implies norm inequality \(\Vert \widetilde{u}\Vert _{W^{1,\Psi }(\Omega )}\le C\Vert {u}\Vert _{Y^{\Psi ,\Phi }(\partial \Omega )}\). If one could find extension operator \(u\mapsto \tilde{u}\) from \(u\) defined on \(\partial \Omega \) to \(\tilde{u}\) defined on \(\Omega \) for which inequality (7.1) holds, it would imply that trace operator from Theorem 2.3, acting from \(W^{1,\Psi }(\Omega )\) to \(Y^{\Psi ,\Phi }(\partial \Omega )\), is a surjection. However we have not proven such property dealing with heat extension operator, our result seems to support that conjecture.
Proof of Theorem 7.1
Using standard covering arguments (see e.g. the book [33], or [34]), suitable partition of the unity on \(\partial \Omega \) and the biLipschitz equivalence of sets \(B(x_0,r)\cap \Omega \), where \(x_0\in \partial \Omega \), \(r\) is sufficiently small, with the cube \(Q=Q'\times (0,1) =(0,1)^n\), we observe that the proof reduces to the case when we deal with the heat extension operator from \(Q'\) to \(Q\). Then we use Lemmas 4.1, 5.1 and 6.2. This requires to verify the condition (4.2) and (5.2).
We start with the verification of (4.2). We have
Let \(G(x,y):= \mathrm{max}\{ x,\Phi (x)\}\). When \(y>1\) we have \(y\prec \Phi (y)\). Consequently \(\mathcal{L}\precsim G(x)\) when \(y<1\) and \(\mathcal{L}\precsim G(x)\Phi (y)\) when \(y>1\). This implies
The verification of the condition \(\int _0^1 G(|\ln t|) dt <\infty \) follows from the following two estimates:
This completes the proof of (4.2). To verify (5.2) we have to check that
This follows from chain of inequalities where \(s<1/2\):
\(\square \)
Our final result establishes regularity properties of solutions to heat equation with the initial condition in weighted Orlicz Slobodetskii space.
Theorem 7.2
Let \(u\in Y^{\Phi ,\Phi }_{L, \log }(\mathbb {R}^{n-1}\times \{ 0\})\), (regularity property) \((\Psi ,\Phi )\) be as in Assumption A (see Definition 2.1), \(\Phi \) satisfy Assumption B (see Definition 6.1), \(T>0\). Moreover, let \(\tilde{u}\in W^{1,1}_{loc}(\mathbb {R}^{n-1}\times (0,T))\) be the solution to heat equation
Then \(\widetilde{u}\in W^{1,\Psi }_L(\mathbb {R}^{n-1}\times (0,T))\) and we have
and constants in the above estimates are independent of \(u\).
Proof
The proof is based on the choice of suitable Lipschitz resolution of unity on \(\mathbb {R}^{n-1}\): \(\{ \phi _i\}_{i\in \mathbb {N}}\) with the control of Lipschitz constants and supports of the \(\phi _i\)’s where \(\mathrm{supp}\phi _i\subseteq Q^{'}_i\) and \(Q_i^{'}\)’s are unit cubes. The we provide the estimates for \(u_i=u\phi _i\). The details are left to the reader. \(\square \)
Example 7.1
The pair \((\Phi (\lambda ),\Psi (\lambda ))= (\lambda (\mathrm{log} (2+\lambda ))^\alpha , \lambda (\mathrm{log} (2+\lambda ))^{\alpha +1})\) where \(\alpha > 0\) obeys assumptions of Theorems 7.1 and 7.2.
Remark 7.2
[Open questions]
-
1.
We do not know what is the optimal space for the initial data \(u\) to have the solution of (7.2) in Orlicz–Sobolev space \(W^{1,\Psi }(\mathbb {R}^{n-1}\times (0,T))\).
-
2.
It would be interesting to know under what conditions one has:\(u\in Y^{\Phi ,\Phi }_{\omega _1}(\Omega )\Rightarrow \tilde{u}\in W^{1,\Psi }_{\omega _2}(\Omega \times (0,T))\) where \(\Omega \) is the given domain and \(\omega _1,\omega _2\) are given measures defined on \(\Omega \) and \(\Omega \times (0,T)\), respectively.
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Acknowledgments
Miroslav Krbec has passed away in 2012, when we were preparing this paper. The time of our cooperation and very nice personality of Miroslav will not be forgotten. The work on the paper started when A.K. visited Institute of Mathematics of Czech Academy of Science in Prague in December 2009. She would like to thank Czech Academy of Science for hospitality.
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A. Kałamajska’s work has been supported by the Polish Ministry of Science Grant no. NN201 397837 (years 2009–2012). M. Krbec’s work was carried out within the framework of the Institutional Research Plan No. AV0Z10190503 of the Institute of Mathematics, Academy of Sciences of the Czech Republic and was supported by the Grant GAČR No. 201/10/1920.
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Kałamajska, A., Krbec, M. Well posedness and regularity for heat equation with the initial condition in weighted Orlicz–Slobodetskii space subordinated to Orlicz space like \(\lambda (\mathrm{log} \lambda )^\alpha \) and the logarithmic weight. Rev Mat Complut 28, 677–713 (2015). https://doi.org/10.1007/s13163-014-0164-4
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DOI: https://doi.org/10.1007/s13163-014-0164-4