Sharp constants for fractional Hardy–Littlewood maximal operators on finite graphs

Abstract For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>α</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0\leq\alpha&lt;1} , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> </m:msub> </m:math> {M_{\alpha,G}} be the fractional Hardy–Littlewood maximal operator on the finite graph G . If <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0&lt;p\leq q\leq 1} , we obtain lower bounds and upper bounds for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>C</m:mi> <m:mrow> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> {C_{G,q}} such that the inequality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mi>f</m:mi> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mi>q</m:mi> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mrow> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mi>f</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> {\|M_{\alpha,G}f\|_{q}\leq C_{G,q}\|f\|_{p}} holds and the complete graphs <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {K_{n}} and star graphs <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {S_{n}} are two extreme graphs that reach the lower bound and upper bound respectively. If <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:mrow> </m:math> {1&lt;p&lt;+\infty} , we prove the existence of the extremizers for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>K</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:msub> <m:mo>∥</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mi mathvariant="normal">ℓ</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>→</m:mo> <m:msup> <m:mi mathvariant="normal">ℓ</m:mi> <m:mi>p</m:mi>