Introduction to Generalized Lebesgue

What is/are Generalized Lebesgue?

Generalized Lebesgue - \;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$\end{document} The study of the problem \begin{document}$ (P_{\lambda}) $\end{document} needs generalized Lebesgue and Sobolev spaces. ^[1] We obtain lower estimates for the generalized Lebesgue constants of Fourier-Jacobi sums, in the spaces of functions being integrable with the $\rho(t) = (1-t)^A (1+t)^B$ weight, therefore confirming an exactness of known upper estimates. ^[2] Finally, we give pointwise convergence of the operators at a generalized Lebesgue point. ^[3] The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. ^[4] We obtain, in some cases, estimates of generalized Lebesgue constants of Fourier-Jacobi sums in $L_{p,w}$ spaces. ^[5] Properties of the generalized Lebesgue constants for the Fourier-Jacobi sums are obtained. ^[6] A global regularity is obtained with respect to the nonhomogeneous term from weighted Lebesgue spaces, Orlicz spaces, and weighted Orlicz spaces, which are generalized Lebesgue spaces, provided that the coefficients have small BMO seminorms and the domains are δ -Reifenberg domains. ^[7] Our analysis utilizes a result from the monotone operator theory and some theory on the generalized Lebesgue and Sobolev spaces. ^[8] By means of these three types of absolute continuity, we establish generalized Egoroff's theorem, generalized Riesz's theorem and generalized Lebesgue's theorem in the framework involving the ordered pair of monotone measures. ^[9] In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(\phi(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }\Omega, u=0 & \ \ \mbox{ on } \partial\Omega, \end{cases}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$), $V$ is a given function in a generalized Lebesgue space $L^{s(x)}(\Omega)$, and $f(x,u)$ is a Caratheodory function satisfying suitable growth conditions. ^[10] Let $$ \mu $$μ a nonnegative Radon measure on $$ {\mathbb{R}}^{d} $$Rd; $$ p,q,\gamma ,k $$p,q,γ,k real numbers; $$ M_{\mu ,k}^{\gamma } $$Mμ,kγ a fractional maximal operator; $$ A_{p,q}^{\gamma ,k} \left( \mu \right) $$Ap,qγ,kμ a Muckenhoupt condition associated to $$ \mu $$μ; $$ L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) $$Lp(·)(Rd,μ) and $$ F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) $$F(q,p,α,μ)(Rd) two generalized Lebesgue spaces. ^[11] \end{aligned}$$Δ(|Δu|p1(x)-2Δu)+Δ(|Δu|p2(x)-2Δu)=λV1(x)|u|q(x)-2u-μV2(x)|u|α(x)-2u,x∈Ωu=Δu=0,x∈∂Ωwhere $$\Omega \in \mathbb {R}^{N}$$Ω∈RN with $$N\ge 2$$N≥2 is a bounded domain with smooth boundary, $$\lambda $$λ, $$\mu $$μ are positive real numbers, $$p_{1}$$p1, $$p_{2}$$p2, q and $$\alpha $$α are continuous functions on $$\overline{\Omega }$$Ω¯, $$V_1$$V1 and $$V_2$$V2 are weight functions in a generalized Lebesgue spaces $$L^{s_1(x)}(\Omega )$$Ls1(x)(Ω) and $$L^{s_2(x)}(\Omega )$$Ls2(x)(Ω) respectively such that $$V_1$$V1 may change sign in $$\Omega $$Ω and $$V_2\ge 0$$V2≥0 on $$\Omega $$Ω. ^[12] We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. ^[13] A considerable number of research has been carried out on the generalized Lebesgue spaces Lp(x) and boundedness of different integral operators therein. ^[14]

Need Generalized Lebesgue

\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right) \end{equation}$\end{document} The study of the problem \begin{document}$ (P_{\lambda}) $\end{document} needs generalized Lebesgue and Sobolev spaces. ^[1] The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. ^[2]

generalized lebesgue space

A global regularity is obtained with respect to the nonhomogeneous term from weighted Lebesgue spaces, Orlicz spaces, and weighted Orlicz spaces, which are generalized Lebesgue spaces, provided that the coefficients have small BMO seminorms and the domains are δ -Reifenberg domains. ^[1] In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(\phi(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }\Omega, u=0 & \ \ \mbox{ on } \partial\Omega, \end{cases}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$), $V$ is a given function in a generalized Lebesgue space $L^{s(x)}(\Omega)$, and $f(x,u)$ is a Caratheodory function satisfying suitable growth conditions. ^[2] Let $$ \mu $$μ a nonnegative Radon measure on $$ {\mathbb{R}}^{d} $$Rd; $$ p,q,\gamma ,k $$p,q,γ,k real numbers; $$ M_{\mu ,k}^{\gamma } $$Mμ,kγ a fractional maximal operator; $$ A_{p,q}^{\gamma ,k} \left( \mu \right) $$Ap,qγ,kμ a Muckenhoupt condition associated to $$ \mu $$μ; $$ L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) $$Lp(·)(Rd,μ) and $$ F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) $$F(q,p,α,μ)(Rd) two generalized Lebesgue spaces. ^[3] \end{aligned}$$Δ(|Δu|p1(x)-2Δu)+Δ(|Δu|p2(x)-2Δu)=λV1(x)|u|q(x)-2u-μV2(x)|u|α(x)-2u,x∈Ωu=Δu=0,x∈∂Ωwhere $$\Omega \in \mathbb {R}^{N}$$Ω∈RN with $$N\ge 2$$N≥2 is a bounded domain with smooth boundary, $$\lambda $$λ, $$\mu $$μ are positive real numbers, $$p_{1}$$p1, $$p_{2}$$p2, q and $$\alpha $$α are continuous functions on $$\overline{\Omega }$$Ω¯, $$V_1$$V1 and $$V_2$$V2 are weight functions in a generalized Lebesgue spaces $$L^{s_1(x)}(\Omega )$$Ls1(x)(Ω) and $$L^{s_2(x)}(\Omega )$$Ls2(x)(Ω) respectively such that $$V_1$$V1 may change sign in $$\Omega $$Ω and $$V_2\ge 0$$V2≥0 on $$\Omega $$Ω. ^[4] A considerable number of research has been carried out on the generalized Lebesgue spaces Lp(x) and boundedness of different integral operators therein. ^[5]

generalized lebesgue constant

We obtain lower estimates for the generalized Lebesgue constants of Fourier-Jacobi sums, in the spaces of functions being integrable with the $\rho(t) = (1-t)^A (1+t)^B$ weight, therefore confirming an exactness of known upper estimates. ^[1] We obtain, in some cases, estimates of generalized Lebesgue constants of Fourier-Jacobi sums in $L_{p,w}$ spaces. ^[2] Properties of the generalized Lebesgue constants for the Fourier-Jacobi sums are obtained. ^[3]