Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

S Samko,E Shargorodsky,B Vakulov

doi:10.1016/j.jmaa.2006.01.069

Abstract

In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229–246], Sobolev-type p ( ⋅ ) → q ( ⋅ ) -theorems were proved for the Riesz potential operator I α in the weighted Lebesgue generalized spaces L p ( ⋅ ) ( R n , ρ ) with the variable exponent p ( x ) and a two-parameter power weight fixed to an arbitrary finite point x 0 and to infinity, under an additional condition relating the weight exponents at x 0 and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L p ( ⋅ ) ( S n , ρ ) on the unit sphere S n in R n + 1 are also improved in the same way.

Highlights

We consider the Riesz potential operator I αf (x) =f (y) |x − y|n−α dy, 0 < α < n, Rn (1.1)in the weighted Lebesgue generalized spaces Lp(·)(Rn, ρ) with a variable exponent p(x) defined by the norm f Lp(·)(Rn,ρ) = inf λ > 0: |f (x)| p(x) ρ (x ) dx 1, λRn where ρ(x) = ργ0,γ∞ (x) = |x|γ0 1 + |x| γ∞−γ0 .We refer to [3,4,5,6] for the basics of the spaces Lp(·) with variable exponent.We assume that the exponent p(x) satisfies the standard conditions (1.2) (1.3)1 < p− p(x) p(x) − p(y) p+ < ∞, x ∈ Rn, A ln 1 |x−y|
The goal of this note is to prove that Theorem 1.1 is valid without the additional condition (1.11)
The spherical potential operator Kα is bounded from the space Lp(·)(Sn, ρβa,βb ) with ρβa,βb (σ ) = |σ − a|βa · |σ − b|βb, where a ∈ Sn and b ∈ Sn are arbitrary points on the unit sphere Sn, a = b, into the space

Summary

Introduction

In the weighted Lebesgue generalized spaces Lp(·)(Rn, ρ) with a variable exponent p(x) defined by the norm f Lp(·)(Rn,ρ) = inf λ > 0:. We refer to [3,4,5,6] for the basics of the spaces Lp(·) with variable exponent. Taken together are equivalent to the following global condition: p(x) − p(y). Q(x) p(x) n x, y ∈ Rn. The following statement was proved in [8]. The goal of this note is to prove that Theorem 1.1 is valid without the additional condition (1.11). We consider a similar statement for the spherical potential operators. F (σ ) |x − σ |n−α dσ, x ∈ Sn, 0 < α < n, Sn in the corresponding weighted spaces Lp(·)(Sn, ρ) on the unit sphere Sn in Rn+1

The case of the spatial potential operator

The case of the spherical potential operator