Nonlocal self-improving properties

Abstract

Solutions to nonlocal equations with measurable coefficients are higher differentiable.\n¶\nSpecifically, we consider nonlocal integrodifferential equations with measurable\ncoefficients whose model is given by\n¶\n<math display="block"><mrow>\n<msub><mrow><mo class="MathClass-op">∫\n</mo></mrow><mrow><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><msub><mrow><mo class="MathClass-op"> ∫\n</mo></mrow><mrow><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mrow><mo class="MathClass-open">[</mo><mrow><mi>u</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow><mo class="MathClass-bin">−</mo><mi>u</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>y</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">]</mo></mrow><mrow><mo class="MathClass-open">[</mo><mrow><mi>η</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow><mo class="MathClass-bin">−</mo><mi>η</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>y</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">]</mo></mrow><mi>K</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi><mo class="MathClass-punc">,</mo><mi>y</mi></mrow><mo class="MathClass-close">)</mo></mrow><mspace class="thinspace" width="0.3em"/><mi>d</mi><mi>x</mi><mspace class="thinspace" width="0.3em"/><mi>d</mi><mi>y</mi>\n<mo class="MathClass-rel">=</mo><msub><mrow><mo class="MathClass-op"> ∫\n</mo></mrow><mrow><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mi>f</mi><mi>η</mi><mspace class="thinspace" width="0.3em"/><mi>d</mi><mi>x</mi><mspace class="quad" width="1em"/><mo class="qopname">for all</mo><mspace class="nbsp" width="1em"/><mi>η</mi>\n<mo class="MathClass-rel">∈</mo> <msubsup><mrow><mi>C</mi></mrow><mrow><mi>c</mi></mrow><mrow><mi>∞</mi></mrow></msubsup><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo class="MathClass-close">)</mo></mrow><mo class="MathClass-punc">,</mo>\n</mrow></math>\n¶\nwhere the kernel\n<math display="inline"><mi>K</mi><mrow><mo class="MathClass-open">(</mo><mrow><mspace class="thinspace" width="0.3em"/>\n<mo class="MathClass-bin">⋅</mo><mspace class="thinspace" width="0.3em"/></mrow><mo class="MathClass-close">)</mo></mrow></math>\nis a measurable function and satisfies the bounds\n¶\n<math display="block"><mrow>\n<mfrac><mrow><mn>1</mn></mrow>\n<mrow><mi>Λ</mi><mo class="MathClass-rel">|</mo><mi>x</mi>\n<mo class="MathClass-bin">−</mo>\n<mi>y</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>n</mi><mo class="MathClass-bin">+</mo><mn>2</mn><mi>α</mi></mrow></msup></mrow></mfrac>\n<mo class="MathClass-rel">≤</mo>\n<mi>K</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi><mo class="MathClass-punc">,</mo><mi>y</mi></mrow><mo class="MathClass-close">)</mo></mrow>\n<mo class="MathClass-rel">≤</mo> <mfrac><mrow><mi>Λ</mi></mrow>\n<mrow><mo class="MathClass-rel">|</mo><mi>x</mi>\n<mo class="MathClass-bin">−</mo>\n<mi>y</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>n</mi><mo class="MathClass-bin">+</mo><mn>2</mn><mi>α</mi></mrow></msup></mrow></mfrac>\n</mrow></math>\n¶\nwith\n<math display="inline"><mn>0</mn>\n<mo class="MathClass-rel"><</mo>\n<mi>α</mi>\n<mo class="MathClass-rel"><</mo> <mn>1</mn></math>,\n<math display="inline"><mi>Λ</mi>\n<mo class="MathClass-rel">></mo> <mn>1</mn></math>, while\n<math display="inline"><mi>f</mi>\n<mo class="MathClass-rel">∈</mo> <msubsup><mrow><mi>L</mi></mrow><mrow><mo class="qopname">loc</mo></mrow><mrow><mi>q</mi></mrow></msubsup><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo class="MathClass-close">)</mo></mrow></math> for\nsome\n<math display="inline"><mi>q</mi>\n<mo class="MathClass-rel">></mo> <mn>2</mn><mi>n</mi><mo class="MathClass-bin">∕</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi>\n<mo class="MathClass-bin">+</mo> <mn>2</mn><mi>α</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>.\nThe main result states that there exists a positive, universal exponent\n<math display="inline"><mi>δ</mi>\n<mo class="MathClass-rel">≡</mo>\n<mi>δ</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi><mo class="MathClass-punc">,</mo><mi>α</mi><mo class="MathClass-punc">,</mo><mi>Λ</mi><mo class="MathClass-punc">,</mo><mi>q</mi></mrow><mo class="MathClass-close">)</mo></mrow></math> such that for every\nweak solution\n[math]\nthe self-improving property\n¶\n<math display="block"><mrow>\n<mi>u</mi>\n<mo class="MathClass-rel">∈</mo> <msup><mrow><mi>W</mi></mrow><mrow><mi>α</mi><mo class="MathClass-punc">,</mo><mn>2</mn></mrow></msup><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo class="MathClass-close">)</mo></mrow><mspace class="quad" width="1em"/><mo class="MathClass-rel">⇒</mo><mspace class="quad" width="1em"/><mi>u</mi>\n<mo class="MathClass-rel">∈</mo> <msubsup><mrow><mi>W</mi></mrow><mrow><mo class="qopname">\nloc</mo></mrow><mrow><mi>α</mi><mo class="MathClass-bin">+</mo><mi>δ</mi><mo class="MathClass-punc">,</mo><mn>2</mn><mo class="MathClass-bin">+</mo><mi>δ</mi></mrow></msubsup><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mo class="MathClass-close">)</mo></mrow>\n</mrow></math>\n¶\nholds. This differentiability improvement is a genuinely nonlocal phenomenon and\ndoes not appear in the local case, where solutions to linear equations in divergence\nform with measurable coefficients are known to be higher integrable but are not, in\ngeneral, higher differentiable.\n¶\nThe result is achieved by proving a new version of the Gehring lemma\ninvolving certain families of lifted reverse Hölder-type inequalities in\n[math] and\nwhich is implied by delicate covering and exit-time arguments. In turn, such reverse\nHölder inequalities are based on the concept of dual pairs, that is, pairs\n[math] of measures and\nfunctions in\n[math]\nwhich are canonically associated to solutions. We also allow for more general equations involving\nas a source term an integrodifferential operator whose kernel does not necessarily have to be\nof order [math] .