Weighted Herz Spaces and Regularity Results

Yuxing Guo,Yinsheng Jiang

doi:10.1155/2012/283730

Abstract

It is proved that, for the nondivergence form elliptic equations <svg style="vertical-align:-6.50204pt;width:114.9625px;" id="M1" height="20.85" version="1.1" viewBox="0 0 114.9625 20.85" width="114.9625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.85)"> <g transform="translate(72,-55.32)"> <text transform="matrix(1,0,0,-1,-71.95,61.83)"> <tspan style="font-size: 12.50px; " x="0" y="0">∑</tspan> </text> <text transform="matrix(1,0,0,-1,-60.52,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> <tspan style="font-size: 8.75px; " x="0" y="9.9300003">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="9.9300003">,</tspan> <tspan style="font-size: 8.75px; " x="4.9098721" y="9.9300003">𝑗</tspan> <tspan style="font-size: 8.75px; " x="8.3143997" y="9.9300003">=</tspan> <tspan style="font-size: 8.75px; " x="14.30952" y="9.9300003">1</tspan> </text> <text transform="matrix(1,0,0,-1,-39.25,61.86)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan> </text> <text transform="matrix(1,0,0,-1,-32.98,58.73)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑖</tspan> <tspan style="font-size: 8.75px; " x="2.7218721" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-25.83,61.86)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> </text> <text transform="matrix(1,0,0,-1,-19.9,58.73)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-15.09,56.54)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑖</tspan> </text> <text transform="matrix(1,0,0,-1,-12.65,58.73)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-7.83,56.54)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-0.56,61.86)"> <tspan style="font-size: 12.50px; " x="0" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="12.040389" y="0">𝑓</tspan> </text> </g> </g> </svg>, if <svg style="vertical-align:-2.34499pt;width:10.675px;" id="M2" height="13.4875" version="1.1" viewBox="0 0 10.675 13.4875" width="10.675" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.4875)"> <g transform="translate(72,-61.21)"> <text transform="matrix(1,0,0,-1,-71.95,63.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑓</tspan> </text> </g> </g> </svg> belongs to the weighted Herz spaces <svg style="vertical-align:-4.27347pt;width:58.799999px;" id="M3" height="17.9" version="1.1" viewBox="0 0 58.799999 17.9" width="58.799999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.9)"> <g transform="translate(72,-57.68)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐾</tspan> </text> <text transform="matrix(1,0,0,-1,-61.42,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑞</tspan> <tspan style="font-size: 8.75px; " x="-1" y="8.7600002">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-56.57,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜑</tspan> <tspan style="font-size: 12.50px; " x="12.490497" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="17.704248" y="0">𝑤</tspan> <tspan style="font-size: 12.50px; " x="27.394073" y="0">)</tspan> </text> </g> </g> </svg>, then <svg style="vertical-align:-6.50204pt;width:105.4375px;" id="M4" height="20.6875" version="1.1" viewBox="0 0 105.4375 20.6875" width="105.4375" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,20.6875)"> <g transform="translate(72,-55.45)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> </text> <text transform="matrix(1,0,0,-1,-66.02,58.86)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-61.21,56.67)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑖</tspan> </text> <text transform="matrix(1,0,0,-1,-58.77,58.86)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑥</tspan> </text> <text transform="matrix(1,0,0,-1,-53.95,56.67)"> <tspan style="font-size: 6.25px; " x="0" y="0">𝑗</tspan> </text> <text transform="matrix(1,0,0,-1,-46.68,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">∈</tspan> <tspan style="font-size: 12.50px; " x="12.027886" y="0">𝐾</tspan> </text> <text transform="matrix(1,0,0,-1,-24.11,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑞</tspan> <tspan style="font-size: 8.75px; " x="-1" y="8.7600002">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-19.26,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜑</tspan> <tspan style="font-size: 12.50px; " x="12.490497" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="17.704248" y="0">𝑤</tspan> <tspan style="font-size: 12.50px; " x="27.394073" y="0">)</tspan> </text> </g> </g> </svg>, where <svg style="vertical-align:-0.11285pt;width:7.5374999px;" id="M5" height="7.1624999" version="1.1" viewBox="0 0 7.5374999 7.1624999" width="7.5374999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,7.1625)"> <g transform="translate(72,-66.27)"> <text transform="matrix(1,0,0,-1,-71.95,66.44)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> </text> </g> </g> </svg> is the <svg style="vertical-align:-0.20064pt;width:32.137501px;" id="M6" height="14.025" version="1.1" viewBox="0 0 32.137501 14.025" width="32.137501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.025)"> <g transform="translate(72,-60.78)"> <text transform="matrix(1,0,0,-1,-71.95,61.03)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑊</tspan> </text> <text transform="matrix(1,0,0,-1,-57.68,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> <tspan style="font-size: 8.75px; " x="4.3759999" y="0">,</tspan> <tspan style="font-size: 8.75px; " x="6.5640001" y="0">𝑝</tspan> </text> </g> </g> </svg>-solution of the equations. In order to obtain this, the authors first establish the weighted boundedness for the commutators of some singular integral operators on <svg style="vertical-align:-4.27347pt;width:58.799999px;" id="M7" height="17.9" version="1.1" viewBox="0 0 58.799999 17.9" width="58.799999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.9)"> <g transform="translate(72,-57.68)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐾</tspan> </text> <text transform="matrix(1,0,0,-1,-61.42,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑞</tspan> <tspan style="font-size: 8.75px; " x="-1" y="8.7600002">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-56.57,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝜑</tspan> <tspan style="font-size: 12.50px; " x="12.490497" y="0">,</tspan> <tspan style="font-size: 12.50px; " x="17.704248" y="0">𝑤</tspan> <tspan style="font-size: 12.50px; " x="27.394073" y="0">)</tspan> </text> </g> </g> </svg>.

Highlights

For a sequence φ {φ k }∞−∞, φ k > 0, we suppose that φ satisfies doubling condition of order s, t and write φ ∈ D s, t if there exists C ≥ 1 such that
1 aij uxixj f, and we prove that if f ∈ Kpq φ, w, uxixj ∈ Kpq φ, w, where u is the W2,p-solution of the equations
Suppose that Ω is an open set of Rn and a ∈ VMO Ω

Summary

Introduction

For a sequence φ {φ k }∞−∞, φ k > 0, we suppose that φ satisfies doubling condition of order s, t and write φ ∈ D s, t if there exists C ≥ 1 such that. Suppose that w is a weight function on Rn. For. Weighted Herz spaces are considered in 5, 6. Lu and Tao in 7 studied nondivergence form elliptic equations on Morrey-Herz spaces, which are more general spaces. We recall continuity results regarding the Calderon-Zygmund singular integral operators that will appear in the representation formula of the uxixj estimates. Throughout this paper, unless otherwise indicated, C will be used to denote a positive constant that is not necessarily the same at each occurrence

Preliminaries

Weighted Boundedness of Commutators

Interior Estimate of Elliptic Equation