Well posedness and regularity for heat equation with the initial condition in weighted Orlicz–Slobodetskii space subordinated to Orlicz space like λ(logλ)α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda (\mathrm{log} \lambda )^\alpha $$\end{document} and the logarithmic weight

We consider the initial-value problem u~t=Δxu~(x,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{u}_t=\Delta _x \tilde{u}(x,t)$$\end{document}, u~(x,0)=u(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{u}(x,0)=u(x)$$\end{document}, where x∈Rn-1,t∈(0,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \mathbb {R}^{n-1},t\in (0,T)$$\end{document} and u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u$$\end{document} belongs to certain weighted Orlicz–Slobodetskii space YlogΦ,Φ(Rn-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y^{ \Phi ,\Phi }_{log}(\mathbb {R}^{n-1})$$\end{document} subordinated to the logarithmic weight. We prove that under certain assumptions on Orlicz function Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi }$$\end{document}, the solution u~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{u}$$\end{document} belongs to Orlicz–Sobolev space W1,Ψ(Ω×(0,T))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,{\Psi }}(\Omega \times (0,T))$$\end{document} for certain function Ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi $$\end{document} which in general dominates Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document}. The typical representants are Φ(λ)=λ(log(2+λ))α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (\lambda )= \lambda (\mathrm{log} (2+\lambda ))^\alpha $$\end{document}, Ψ(λ)=λ(log(2+λ))α+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi (\lambda )= \lambda (\mathrm{log} (2+\lambda ))^{\alpha +1}$$\end{document} where α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > 0$$\end{document}.


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 , log ( ), is a N -function. It consists of all v ∈ L ( ) (the Orlicz space generated by ), for which the seminorm is finite. As our main result formulated in Theorem 7.2, we prove that if satisfies certain assumptions (Assumption B in Definition 6.1), then the solutionũ of (1.1) lies in the Orlicz-Sobolev space W 1, (R n−1 × (0, T )), i. e.ũ, together with its all first order partial derivatives belongs to L (R n−1 × (0, T )), where is in a sense conjugate to (see Definition 2.1). The natural representative pair of admitted functions would be functions generating the logarithmic Zygmund spaces: (λ) = λ(ln(2 + λ)) α and (λ) = λ(ln(2 + λ)) α+1 , where α > 0. They cannot grow to fast. The conjugate of is equivalent to exp t 1/α near infinity and does not satisfy the 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 α,q p ( ) we refer to papers: [20,43,51,[58][59][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 is essentially different than λ 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 solutionũ of (1.1) lies in the Orlicz-Sobolev space W 1,R ( × (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 and . This motivated us to consider weighted Orlicz-Slobodetskii setting, while our previous analysis did not require weights.
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 ofũ 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. loc (R) [52]. We will be also dealing with the time-directional maximal function of function w ∈ L 1 loc (R n−1 × [0, ∞)), the function (defined for almost every (x, t 0 )). Having to norms · and · 1 defined on a Banach space X , we will write · ∼ · 1 if norm · is equivalent to · 1 on X . Having two functions , defined on [0, ∞) we will say that dominates ( ≺ ) if there exist constants C 1 , Functions , are called equivalent if ≺ and ≺ . The notation " " will be used in usual manner, namely, if , : A → R are given functions, where 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 if there is a constant C > 0 such that (a) ≤ C (a), for every a ∈ A. When n ∈ N, we denote: Q = [0, 1] n−1 , Q = [0, 1] n = Q × (0, 1). By Li p( ) we denote space of Lipschitz functions defined on the set ⊆ R n , while by Li p 0 ( ) we denote those elements of Li p( ) which have compact support in . If x = (x 1 , . . . , x n ) ∈ R n , then x will stand for (x 1 , . . . , x n−1 ) ∈ R n−1 . By ln x we denote the natural logarithm of a positive number x.

Orlicz space
When : [0, ∞) → [0, ∞) is a nondecreasing convex continuous function such that (0) = 0 and lim t→∞ (t) = +∞, 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 (λ) = λ p and p ≥ 1, then L ( ) = L p ( ) is the usual Lebesgue space. The same notation will be used for vector functions, u : → 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 ∈ 2 if it satisfies the 2 -condition: (2λ) ≤ C (λ), for every λ > 0, with a constant C independent of λ. Symbol ∈ c 2 will mean that the Legendre conjugate of , that is, * (s) := sup t>0 {st − (t)}, satisfies the 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, μ) be the measurable space equipped with the measure μ. Then the expression f L (X,μ),α := inf λ > 0 : defines a complete norm on Here D α u means the distributional derivative of u. We define the space W k, ∞ ( ), (respectively W k, L ( )) as the completion of C ∞ (¯ ) (respectively Li p( )) in the norm of the space W k, ( ).

Orlicz-Slobodetskii space Y ,
Let ⊆ R n be an open bounded domain, , : [0, ∞) → [0, ∞) be nondecreasing convex continuous functions such that (0) = (0) = 0 and lim t→∞ (t) = lim t→∞ (t) = +∞. By Y , ( ) we denote the space of all u ∈ L ( ), for which the seminorm is finite. We equip it with the norm: is the Luxemburg-type seminorm. Analogously one can define Y , (u, M), I (u, M), and J (u, M), where M ⊆ R k is an arbitrary n-dimensional rectifiable set (n ≤ k) and instead of the Lebesgue measure we consider the n-dimensional Hausdorff measure H n defined on M.
By Y , ∞ ( ) (respectively Y , L ( )) we will mean the completion of set which is the norm of u in the Slobodetskii space W 1− 1 p , p (∂ ), see e.g. [33].

Basic assumptions
In the sequel we will be dealing with the following assumptions comming from papers by Kita [27,28].

Trace operator
Let us briefly recall basic claims from [23, Theorems 3.10 and 3.13]. The original formulation holds with u ∈ C ∞ (¯ ) however, the proof follows by the same arguments with no difference for u ∈ Li p( ) as well.
with constants C, C 1 , C 2 independent of u. (ii) If (2.4) holds with some s 0 > 0, then for every u ∈ Li p( ), with constants C, C 1 , C 2 independent of u.

Trace operator
Let the assumptions in Theorems 2.1 and 2.2 be satisfied and let u ∈ W 1, Li p ( ). Consider any sequence u m ∈ L(¯ ) convergent to u in the norm of the space W 1, ( ). Then {u m } is a Cauchy sequence in Y , (∂ ) (convergence in the norm), so that it converges some elementū ∈ Y , L (∂ ). It is easy to observe thatū is independent of the choice of the sequence {u m } ⊆ L(¯ ), converging to u. It allows to extend the standard definition of the trace operator: where the convergence holds in the norm of Y , (∂ ).
As a consequence we obtain the following result. (2.10)

Heat kernel estimates
Let u ∈ Li p(∂ ). We will define function u ∈ Li p( ) such that Tr u = u using the Gaussian kernel. Let be the heat kernel. Then E obeys the following properties 1.
We will start our construction with the case when = Q, and assuming that u ∈ Li p 0 (Q ).
We define where g * u is the usual convolution. We have the following observation. The remaining estimates and our final result will be established in several steps presented in in the sequel.

Presentation of main results
We start with the following result.
Our goal is to show that in some cases the function u can have better integrability properties than u. As main result of this section we obtain the following lemma. Lemma 4.1 Let u be given by (3.1), where u ∈ Li p 0 (Q ). Moreover, let ( , ) be as in Assumption A (see Definition 2.1) and suppose that satisfies the following estimate:

2)
and continuous, nondecreasing function G such that where constant C > 0 above is independent of u. Then Indeed, this follows from the following estimates:

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.

Lemma 4.2 Let β > −2 and
Then there exist constants C 0 , C 1 (depending on β) such that for every y ∈ R + we have Proof Function F β is increasing up to certain x β and it is bounded. Therefore inequality follows with Then for any s, t 0 , r > 0 we have Proof We start with the case 0 < r < t 0 2 . Let us find the negative integer k ≤ −2 such In case t 0 2 ≤ r < t 0 , we have (we change variables, substituting q = s 2 4t ) Finally, if r ≥ t 0 , we proceed similarly as above, to get then we use Lemma 4.2 (note that s 2 8r ≤ s 2 8t 0 =: y) to get in this case Therefore the estimate holds in all considered cases.
Now it suffices to apply Lemma 4.3.

Remark 4.2
We observe that under the notation in Lemma 4.4 we have Lemma 4.4. Then for any convex function R and every t 0 > 0 we have Integrating the above expression over R n−1 , then applying Fubini's theorem, we get This finishes the proof of the lemma.
Proof If x ∈ Q and y ∈ Q , then |x − y| ≤ √ n − 1. We have It suffices to show that where C 0 is independent in u. We start with the case t < 1 64 . For this, let We have Let us fix x = (x 2 , . . . , x n−1 ) ∈ Q and consider the mapping We will show that f 1 (x 1 ) takes its minimal value at Define for s ∈ (0, 1): Note that as for s 1 As we have By an obvious modification of the above argument we have and right hand side is nonzero provided that 2 √ t < |(1/2, . . . , 1/2)| = √ 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 ω n−2 is measure of unit sphere in R n−2 is the case n > 2 and ω 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 > 1 64 the computations become simpler as then we have We are now in position to prove Lemma 4.1.
Proof of Lemma 4.1 Let λ > 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 = , 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 C 1 + Q ( |u(x)| λ ) 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 λ = u L (Q ) .

Presentation of main results
Our approach will be based on the following results.
Proposition 5.2 ([23]) For any convex function R, any 0 < ε < 1 2 and any function u ∈ Li p 0 (Q ) such that dist(supp u, ∂ Q ) ≥ ε we have In particular, when ∇ x u 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.
where ω(x, y) = 1 |x−y| . This implies the estimates in the norms where Y R,R ω (Q ) is weighted Orlicz-Slobodetskii space introduced in [11,12]. In particular more precise statement than that in Corollary 5.1 holds.

Remark 5.3
Constant bounds in the above estimations depend on > 0.

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.
Proof The proof is the little modification of Lemma 4.3 with the difference that instead of E α (s, t) we now deal with E α,γ (s, t) = E α (s, t)( s 2 4t ) γ . Repeating the proof of Lemma 4.3 we observe that Using Lemma 4.2 dealing with y = s 2 8t 0 and β = α + γ − 2 > −2, we obtain that with some positive constant C 2 > 0 depending on α + γ . This completes the proof.

Lemma 5.3
Let u ∈ Li p 0 (Q ) and u be given by (3.1). Then for any t 0 > 0 we have where constant C 2 > 0 does not depend on u.

Lemma 5.4 Let i ∈ {1, 2, 3} and (P i u)(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 < < 1/2 be given and u ∈ Li p 0 (Q ) be such that dist(suppu, ∂ Q ) ≥ . 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 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 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 := 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) on R involvingx = C 1 ,ȳ = |u(y)−u(x)| |x−y| . The proof of the estimate of P 1 goes more or less along the lines of that of Lemma 4.5. Namely, we observe that for x ∈ Q Now it suffices to verify that for small s, 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 , therefore if we enlarge C, then we get R((P 3 u)) ≤ 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) (1 + | ln t|), for every t ∈ (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: Letx := C(1 + | ln t|),ȳ := |u(x)−u(y)| |x−y| . Using the condition (5.5) on R(xȳ) and estimating further, we get When T ≥ t ≥ C 0 , the estimates become simples as then we have |x−y| n−2 dy ∼ 1 (as x ∈ Q ). Therefore and by Jensen's inequality After integrating it over (C 0 , T ) × Q and using (5.5) again, we obtain which finishes the proof.
Proof of Lemma 5. 1 We note that for an arbitrary λ > 0 we have by Theorem 4.1, dxdt, hence L 1 ≺ 1 + L 2 and it remains to estimate the expression L 2 . According to Lemma 5.3 we have (with the same notation) Applying Lemma 5.4 with R = and observing that ( 4 j=1 a j ) ≤ 1 4 4 j=1 (4a j ) 1 + 4 j=1 (a j ), we arrive at This gives (5.3). The choice of which together with Proposition 2.1 gives (5.4). This ends the proof of the lemma.
6 Estimates of function ∂ t u

Presentation of results and discussion
Our goal here is to continue our estimates of heat extension operator, dealing now with the time derivative of u.
We start with the following lemma. The Lemma was originally proven for u ∈ C ∞ 0 (Q ). We remark that assumption u ∈ Li p 0 (Q ) does not change final conclusion and the proof under such assumption follows by the same arguments. Lemma 6.1 ([23]) Let u ∈ Li p 0 (Q ), u be given by (3.1), 0 < ε < 1 2 and let dist(supp u, ∂ Q ) ≥ ε. Then we have 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.
We obtained the following result (formulated originally for u ∈ C ∞ 0 (Q )).

Proposition 6.1 ([23]) If R satisfies an Assumption B (see Definition 6.1), then for any function u ∈ Li p 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 It is clear from Proposition 6.1 that under certain assumptions on function R, condition u ∈ Y R,R L (Q ) implies ∂ t u ∈ L R (Q). Following our previous schema, we would like to prove that the condition u ∈ Y , (Q ) implies ∂ t u ∈ L (Q), where ( , ) 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 < < 1/2, u ∈ Li p 0 (Q ) is such that dist(suppu, ∂ Q ) ≥ and u is given by (3.1). Moreover, let ( , ) be as in Assumption A (see Definition 2.1) and satisfies Assumption B (see Definition 6.1). Then we have for every λ > 0:

Remark 6.3
Note that under the assumptions of Lemma 6.2 we have and parameter δ > 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 δ = 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.
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 E α,w (s, t) = E α, 1 estimates for E α, 1 2 (s, t) and E α, 3 2 (s, t), after we note that F β (x) 1 for large arguments, so that F β ( s 2 4t 0 ) can be estimated by constant when t 0 < s 2 4C 3 . On the other hand F β (x) ∼ x for small x. Lemma 6.4 Let u ∈ Li p 0 (Q ) and 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) = E n 2 ,v (s, t). Therefore the result follows from Lemma 6.3 applied with α = n 2 .
We are now to establish our crucial estimates for ∂ t u. We have the following result.
Lemma 6.5 Let u ∈ Li p 0 (Q ) and u be as in Lemma 6.2, i ∈ {1, 2, 3} and (C i u)(x, t) be the same as in Lemma 6.4 and R satisfies Assumption B (see Definition 6.1). Then we have
Step 1 (proof of 6.5). The estimate for i = 1 is a consequence of Remark 6.2 as 3 2 t).
For this purpose we note that Step 4 (proof of (6.6) and (6.7) with B 2 u instead of C 2 u).
For this purpose we apply inequality which is obtained by similar arguments as the ones to get (6.9). Observing that P( ch √ t ) P(h), we get, by similar computations as in previous step that This easily gives the desired estimates.
To this goal we again use the fact that when t is sufficiently small, i.e. t < t 0 for some t 0 ∈ (0, 1), we have Now we estimate the integral in brackets {·} denoted by Y . Note that on set of integration we have 1 |x−y| 1 √ t , moreover, P(λ) λ ε for arbitrary ε > 0 (see Remark 6.1). Taking this into account we get: where β = 2(1 − α − ε) > 0, (it is enough to take sufficiently small ε). This implies I 1 1.
To estimate the term with B we note that on set of integration we have P( h √ t ) P(h), so that The estimates when t > t 0 become simpler as on set if integration we have √ 4C 3 t 0 < |x − y|, therefore we omit them. Lemma is proved.
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.

Final results
We are now to present our main results. For this purpose we introduce the new space of functions. Let ⊆ R n be a Lipschitz boundary domain. By Y , log (∂ ) we will mean the modification of the space Y , (∂ ), where the seminorm: is substituted by where σ is the n − 1-dimensional Hausdorff measure on ∂ . By Y , L ,log (∂ ) we will mean the completion of set {u ∈ Li p(∂ ) ∩ Y , log (∂ )} in the norm of Y , log (∂ ). Our first final result reads as follows. with an arbitrary δ > 0 independent on u. Note that ∂ (|u|)dσ (x) ≺ 1 + ∂ (|u|)dσ (x), so this inequality is very close to the following one: and the above implies norm inequality u W 1, ( ) ≤ C u Y , (∂ ) . If one could find extension operator u →ũ from u defined on ∂ toũ defined on for which inequality (7.1) holds, it would imply that trace operator from Theorem 2.3, acting from W 1, ( ) to Y , (∂ ), 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 ∂ and the biLipschitz equivalence of sets B(x 0 , r ) ∩ , where x 0 ∈ ∂ , r is sufficiently small, with the cube Q = Q × (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 (x y) = x y P(x y) x y(1 + P(x) + P(y)) = x y + y (x) + x (y) =: L.
The verification of the condition 1 0 G(| ln t|)dt < ∞ follows from the following two estimates: 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, (R n−1 × (0, T )). 2. It would be interesting to know under what conditions one has:u ∈ Y , ω 1 ( ) ⇒ u ∈ W 1, ω 2 ( ×(0, T )) where is the given domain and ω 1 , ω 2 are given measures defined on and × (0, T ), respectively.