A weak inequality in fractional homogeneous Sobolev spaces

2023/12/22 by Lifeng Wang, Wang, Lifeng
#math.CA

paper · pdf · doi:10.48550/arxiv.2312.14662

Abstract

In this paper, we prove the following inequality ‖(∫_ℝⁿ\frac|f(⋅+y)-f(⋅)|^q|y|^n+sqdy)^(1)/(q)‖_{L^p,∞(ℝⁿ)}\lesssim‖f‖_L^p_s(ℝⁿ), where ‖⋅‖_{L^p,∞(ℝⁿ)} is the weak L^p quasinorm and ‖⋅‖_L^p_s(ℝⁿ) is the homogeneous Sobolev norm, and parameters satisfy the condition that 1<p<q, 2≤ q<∞, and 0<s=n((1)/(p)-(1)/(q))<1. Furthermore, we prove the estimate ‖\mathfrakg_s,q(f)‖_L^p(ℝⁿ)\lesssim‖f‖_F^s_p,q(ℝⁿ) when 0<p,q<∞, -1<s<1, ‖⋅‖_F^s_p,q(ℝⁿ) denotes the homogeneous Triebel-Lizorkin quasinorm and the Littlewood-Paley-Poisson function \mathfrakg_s,q(f)(⋅) is a generalization of the classical Littlewood-Paley g-function. Moreover, we prove the weak type (p,p) boundedness of the G_λ,q-function and the R_s,q-function, where the G_λ,q-function is a generalization of the well-known classical Littlewood-Paley g_λ^*-function. We also prove that when 0<p,q<∞ and -∞<s≤max\0,n((1)/(p)-(1)/(q))\, we have ‖(∫_ℝⁿ\frac|f(⋅+y)-f(⋅)|^q|y|^n+sqdy)^(1)/(q)‖_L^p(ℝⁿ)=∞.

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