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Existence and uniqueness results for solutions of nonlinear equations with right hand side in L 1

A. Fiorenza;  C. Sbordone — 1998

Studia Mathematica

We prove an existence and uniqueness theorem for the elliptic Dirichlet problem for the equation div a(x,∇u) = f in a planar domain Ω. Here f ∈ L 1 ( Ω ) and the solution belongs to the so-called grand Sobolev space W 0 1 , 2 ) ( Ω ) . This is the proper space when the right hand side is assumed to be only L 1 -integrable. In particular, we obtain the exponential integrability of the solution, which in the linear case was previously proved by Brezis-Merle and Chanillo-Li.

A New Proof of the Boundedness of Maximal Operators on Variable Lebesgue Spaces

D. Cruz-Uribe;  L. Diening;  A. Fiorenza — 2009

Bollettino dell'Unione Matematica Italiana

We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space L p ⁢ ( ⋅ ) whenever the exponent function p ⁢ ( ⋅ ) satisfies log-Hölder continuity conditions. We include the case where p ⁢ ( ⋅ ) assumes the value infinity. The same proof also shows that the fractional maximal operator M a , 0 < a < n , maps L p ⁢ ( ⋅ ) into L q ⁢ ( ⋅ ) , where 1 / p ⁢ ( ⋅ ) - 1 / q ⁢ ( ⋅ ) = a / n .

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