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Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier

Troels Roussau Johansen (2011)

Studia Mathematica

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) S R associated with the Jacobi transform through the defining relation S R f ^ ( λ ) = 1 | λ | R f ̂ ( t ) for a function f on ℝ is shown to be bounded from L p ( , d μ ) into L p ( , d μ ) + L ² ( , d μ ) for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from L p , 1 ( , d μ ) into L p , ( , d μ ) + L ² ( , d μ ) . In particular S R f ( t ) R > 0 converges almost everywhere towards f, for f L p ( , d μ ) , whenever (4α + 4)/(2α + 3) < p ≤ 2.

Almost global solutions for non hamiltonian semi-linear Klein-Gordon equations on compact revolution hypersurfaces

Jean-Marc Delort, Jérémie Szeftel (2005)

Journées Équations aux dérivées partielles

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.

Almost homoclinic solutions for a certain class of mixed type functional differential equations

Joanna Janczewska (2011)

Annales Polonici Mathematici

We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: q ̈ ( t ) + V q ( t , q ( t ) ) + u ( t , q ( t ) , q ( t - T ) , q ( t + T ) ) = f ( t ) , where t ∈ ℝ, q ∈ ℝⁿ and T>0 is a fixed positive number. By an almost homoclinic solution (to 0) we mean one that joins 0 to itself and q ≡ 0 may not be a stationary point. We assume that V and u are T-periodic with respect to the time variable, V is C¹-smooth and u is continuous. Moreover, f is non-zero, bounded, continuous and square-integrable....

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