Riesz potentials of Hardy-Hausdorff spaces and Q -type spaces

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https://doi.org/10.1007/s11425-018-9443-7

Riesz potentials of Hardy-Hausdorff spaces and Q-type spaces Pengtao Li School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China Email: [email protected] Received June 21, 2018; accepted September 2, 2018

Abstract

This study investigates the restriction problem for the Riesz potentials of Hardy-Hausdorff spaces

1 (Rn ) and Q-type spaces Q (Rn ). By exploiting a geometric-measure theory generated by the indicatorHH−γ γ 1 (Rn ) into the like functions of compact sets, it is proved that the Riesz operator Iα continuously maps HH−γ

weak Morrey spaces Lq,λ µ,∗ induced by a Radon measure µ, which obeys a geometric condition. Keywords MSC(2010)

restriction theorem, Riesz potential, Hardy-Hausdorff space, Q-type space 31C15, 42B35, 46E35

Citation: Li P T. Riesz potentials of Hardy-Hausdorff spaces and Q-type spaces. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-9443-7

1

Introduction

The restriction problem for the Riesz potentials of function spaces is a critical subject in geometric potential analysis. The Riesz potentials can be applied to reflect the local behavior of functions and distributions. Therefore, over the last several decades, Riesz potentials have been applied extensively to the study of second-order elliptic differential equations and incompressible fluid equations (see, for example, [13, 14, 18, 20, 22, 27, 30, 31]). For f ∈ S (Rn ), let ∫ fb(ξ) = e−ixξ f (x)dx Rn

be the Fourier transform of f at ξ. For s ∈ R, the notation (−∆)α/2 is the α/2-th power of the Laplacian −∆ = −

n ∑ j=1

∂j2 = −

n ∑ ∂2 ∂x2j j=1

c determined by the partial derivatives {∂j = ∂/∂xj }nj=1 and the Fourier transform (·): \ α/2 f (ξ) = |ξ|α fb(ξ). (−∆)

(1.1)

When α > 0, Iα := (−∆)−α/2 is usually called the Riesz operator or the fractional integral. Let α ∈ (0, n). The Riesz operator Iα acting on a Lebesgue measurable function f in Rn is defined by ∫ Iα f (x) = |y − x|α−n f (y)dy Rn

c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝

math.scichina.com

link.springer.com

2

Sci China Math

Li P T

whose constant multiple

( u=

) Γ( (n−α) ) 2 Iα f n π 2 2α Γ( α2 )

may be considered as a solution of the α/2-th order Laplace equation (−∆)α/2 u = f in terms of the Fourier transform. On Hardy spaces and their generalizations, the restriction problem for Riesz potentials has been investigated by several studies. For the Hardy space H 1 (Rn ), by aid of the Poisson integral and the boundary value of harmonic functions, Stein and Weiss [24] obtained the following embedding: Iα H 1 (Rn ) ⊂ Ln/(n−α) (Rn ),

α ∈ (0, n).

(1.2)

Let β ∈ [0, n] and µ be a non-negative Radon measure on Rn . The β-th order ball variation of µ is denoted by |||µ|||β = sup r−β µ(B(x, r)). (1.3) (x,r)∈Rn ×(0,∞)

By exploiting the Riesz potential Iα (1K ), where 1K denotes the characteristic function of a compact set K, Xiao [32] obtained the following restriction result for the Riesz potentials of the associate Morrey spaces H p,