On Integral Inequalities on the Set of Functions with some Properties of Monotonicity

Abstract

Hölder, Jensen and Hardy inequalities play a very important role in the mathematical analysis and in the theory of differential equations. These inequalities are exact on the set of all measurable (non-negative) functions (see book [1]: the results on Hardy inequalities, obtained by B. Muckenhoupt, V. Kokilashvili, J. Bredly, V. Maz’ya, A. Rosin and others). But, in the field of mathematical analysis and, first of all, in the theory of function spaces it is often necessary to use such estimates on sets of non-negative functions with additional properties of monotonicity. On these sets such inequalities lose their exactness, that prevent obtaining the exact results on the basis of them. Let us remind that the most important examples of such functions on R + = (0, ∞) are distribution functions, non-increasing rearrangements and moduli of continuity in the theory of function spaces: % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS % baaSqaaiaadAgaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0JaamyB % aiaadwgacaWGZbWaaiqaaeaadaGacaqaaiaadIhacaGG6aWaaqqaae % aadaabcaqaaiaadAgacaGGOaGaamiEaiaacMcaaiaawIa7aiabg6da % +iaadshaaiaawEa7aaGaayzFaaaacaGL7baacaGGSaGaamOzamaaCa % aaleqabaGaaiOkaaaakiaacIcacaWG1bGaaiykaiabg2da9iGacMga % caGGUbGaaiOzamaaceaabaWaaiGaaeaacaWG0bGaeyyzImRaaGimai % aacQdacqaH7oaBdaWgaaWcbaGaamOzaaqabaGccaGGOaGaamiDaiaa % cMcacqGHKjYOcaWG1baacaGL9baaaiaawUhaaiaacUdaaaa!62FB! $${\lambda _f}(t) = mes\left\{ {\left. {x:\left| {\left. {f(x)} \right| > t} \right.} \right\}} \right.,{f^*}(u) = \inf \left\{ {\left. {t \geqslant 0:{\lambda _f}(t) \leqslant u} \right\}} \right.;$$ % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCa % aaleqabaGaaiOkaiaacQcaaaGccaGGOaGaamyDaiaacMcacqGH9aqp % daWcaaqaaiaaigdaaeaacaWG1baaamaapedabaGaamOzamaaCaaale % qabaGaaiOkaaaakiaacIcacaWG2bGaaiykaiaadsgacaWG2bGaai4o % aaWcbaGaaGimaaqaaiaadwhaa0Gaey4kIipakiabeM8a3naaDaaale % aacaWGRbaabaGaamiCaaaakiaacIcacaWGMbGaaiilaiaadwhacaGG % PaGaeyypa0tbaeqabiqaaaqaaiGacohacaGG1bGaaiiCaaqaamaaee % aabaWaaqGaaeaacaWGObaacaGLiWoacqGHKjYOcaWG1baacaGLhWoa % aaGaaiiFaiaacYhacqqHuoardaqhaaWcbaGaamiAaaqaaiaadUgaaa % GccaWGMbGaaiiFaiaacYhadaWgaaWcbaGaamitamaaBaaameaacaWG % WbaabeaaaSqabaGccaGG7aaaaa!662D! $${f^{**}}(u) = \frac{1}{u}\int_0^u {{f^*}(v)dv;} \omega _k^p(f,u) = \begin{array}{*{20}{c}} {\sup } \\ {\left| {\left. h \right| \leqslant u} \right.} \end{array}||\Delta _h^kf|{|_{{L_p}}};$$ and Peetre’s K— functional in the interpolation theory % MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaacI % cacaWGMbGaaiilaiaadwhacaGGPaGaeyypa0tbaeqabiqaaaqaaiGa % cMgacaGGUbGaaiOzaaqaaiaadAgadaWgaaWcbaGaaGimaaqabaGccq % GHRaWkcaWGMbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamOzaaaa % caGGOaGaaiiFaiaacYhacaWGMbWaaSbaaSqaaiaaicdaaeqaaOGaai % iFaiaacYhacaWGgbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaamyD % aiaacYhacaGG8bGaamOzamaaBaaaleaacaaIXaaabeaakiaacYhaca % GG8bGaamOramaaBaaaleaacaaIXaaabeaakiaacMcacaGGSaGaamOz % aiabgIGiolaadAeadaWgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGgb % WaaSbaaSqaaiaaigdaaeqaaaaa!5EEC! $$K(f,u) = \begin{array}{*{20}{c}} {\inf } \\ {{f_0} + {f_1} = f} \end{array}(||{f_0}||{F_0} + u||{f_1}||{F_1}),f \in {F_0} + {F_1}$$