The rate of convergence on fractional power dissipative operator on some sobolev type spaces

In [3], Chen, Deng, Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces Lp(ℝn), Hardy spaces Hp(ℝn) and general mixed norm spaces, which implies almost everywhere convergence of such operator. In this paper, we study the rate of convergence on fractional power dissipative operator on some sobolev type spaces.


§1 Introduction
We consider the fractional power dissipative equation where n ≥ 2, α > 0, f : R n → C is the given initial data, and ∆ = where f is the Fourier transform of f , and f ∨ is the inverse Fourier transform of f . We can also write u(x, t) as a convolution operator: ), In [10], Miao, Yuan and Zhang proved that K α (x) satisfies that for all α > 0, which immediately implies e −t(−∆) α f is bounded on Lebesgue spaces L p (R n ) for 1 ≤ p < ∞. Also, by the multiplier theorem of Calderón and Torchinsky [1], e −t(−∆) α f is bounded on Hardy spaces H p (R n ) for all 0 < p < ∞. We can see [3] on the boundedness of e −t(−∆) α f on other spaces. Also we can see [6,[8][9][10][11]15] on the fractional power dissipative equations with different potentials. For λ > 0, let I −λ denote the Riesz potential, i.e.
Let H p (R n ) denote the Hardy space, which is defined by where P is the Poisson kernel. A basic result is that L p (R n ) and H p (R n ) are equivalent when p > 1. We further introduce the atomic decomposition: if g ∈ H p (R n ), then where each a j is a (p, 2)-atom. Here a is a (p, 2)-atom if there exists a cube Q such that (1) a is supported in Q, means the largest integer that is not more than x. In the following argument, we mainly discuss the Sobolev type spaces For such function f , we define This space is the classical homogeneous Sobolev spaces for p ≥ 1 and the Hardy-Sobolev spaces for 0 < p ≤ 1 (see [7,14]).
We have already known the fractional power dissipative operator and related maximal operator are bounded on H p (R n ), which implies for each f ∈ H p (R n ), lim t→0 e −t(−∆) α f = f a.e. Then we ask: if f has more regularity, can we obtain better estimates? Our result is the following.
Firstly we consider the case for λ = 0, by the method from [13], Theorem 1.1 can be reduced to This result is correct since the fractional power dissipative operator is bounded on H p (R n ) and holds for all 0 < p < ∞. From now on, let 0 < λ < 2α. We repeat the method of [13], then Theorem 1.1 can be reduced to the following result.
We will prove this theorem in section 2. We decompose the integral in Theorem 1.2 into two parts. As the argument in Chen [2], the estimate of kernel on each part is core of our proof, which is left at Lemma 2.2.
Throughout this paper, A B means that there exists a constant C > 0 independent of all essential variables such that A ≤ CB, and A ∼ B means A B and B A. The space of all infinitely differentiable functions on R n is denoted by There is a standard result of Stein [12] on H p (R n ). We state it here. Lemma 2.1. [12] Let 0 < p ≤ 1. Suppose that a function ζ vanishes at ∞ and satisfies where To prove Theorem 1.2, it suffices to show Set h = I −λ (f ), the above inequality is reduced to Let ϕ 0 , ϕ ∞ ∈ C ∞ (R n ) be radial functions satisfying the following conditions: and ϕ 0 + ϕ ∞ = 1. Then we divide the integral g α,λ to g α,λ 0 and g α,λ ∞ : Firstly we consider g α,λ 0 . Via the similar method of Lemma 2.1 in [4] (we also can see [5]), one concludes: for 0 < λ < 2α, all β = (β 1 , ..., β n ) with β j ≥ 0 for 0 ≤ j ≤ n, When p > 1, by (1), which is controlled by one integrable radially decreasing function due to 0 < λ < 2α. Therefore Here we use that L p is equivalent to H p when p > 1. When 0 < p ≤ 1, by (1), Next we consider g α,λ ∞ . We have the following estimate (proof can be seen in the following Lemma 2.2): (2) holds for all L ≥ 0 so that n + |β| − λ + L ≥ 0. When p > 1, by (2), g α,λ ∞ is controlled by one integrable radially decreasing function, then When 0 < p ≤ 1, by the atomic decomposition of h, h = ∑ j λ j a j , where each a j is a (p, 2)-atom and ∥h∥ p H p ∼ ∑ j |λ j | p . The properties of atom can be seen in section 1. Then So if for each atom a, we have We prove (3) in the last lemma. The proof of Theorem 1.2 is completed as all cases have been proved.
We only estimate the second term that is denoted by K λ (x), then the first term can be treated by the same way.
As to K λ (x), we decompose it by the partition of unity, where η and δ are all in Noting the support of ϕ ∞ , we have We claim: In fact, write Using the support of δ(2 −j ·), we obtain There is a basic fact: for γ = (γ 1 , ..., γ n ), holds for all x ̸ = 0. Since γ is arbitrary, we take the supremum over all γ with |γ| = M , then (5) implies (4).
Proof: We assume that supp a ⊂ Q, where Q is a cube with the center of zero. Other cases are similar since this operator is a convolution operator (it commutes with translations). Then For I, since g α,λ ∞ is controlled by one integrable radially decreasing function, T (f ) = sup t>0 |f * (g α,λ ∞ ) t | is bounded on L 2 . Then Next we start to estimate II. Set N = [n( 1 p − 1)]. Considering the vanishing property of a, for some θ ∈ (0, 1) such that