Let (Tt)t>0 be a symmetric contraction semigroup on the spacesLp(M) (1 ≤ p #x2264; ∞), and let the functions ϕ and ψ be “regularly related”.We show that (Tt)t>0 is ϕ-ultracontractive, i. e., that (Tt)t>0 satisfies the condition‖Ttƒ‖∞ #x2264; Cϕ(t)-l‖ƒ‖p for all ƒ in Lp(M) andall t in R+, if and only if the infinitesimal generator S has Sobolev embedding properties, namely,‖ ψ-αƒq #x2264; C‖f‖p for all f in LP(M), whenever 1 < p < q <∞ and. We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on m for m to map LP(M) to U(M), and for the example where there exists ß in R+ such that)(t)= for all in R+, we give conditions which ensure that the maximal function suptaTtf is bounded
Cowling, M., Meda, S. (1993). Harmonic analysis and ultracontractivity. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 340(2), 733-752 [10.1090/S0002-9947-1993-1127154-7].
Harmonic analysis and ultracontractivity
MEDA, STEFANO
1993
Abstract
Let (Tt)t>0 be a symmetric contraction semigroup on the spacesLp(M) (1 ≤ p #x2264; ∞), and let the functions ϕ and ψ be “regularly related”.We show that (Tt)t>0 is ϕ-ultracontractive, i. e., that (Tt)t>0 satisfies the condition‖Ttƒ‖∞ #x2264; Cϕ(t)-l‖ƒ‖p for all ƒ in Lp(M) andall t in R+, if and only if the infinitesimal generator S has Sobolev embedding properties, namely,‖ ψ-αƒq #x2264; C‖f‖p for all f in LP(M), whenever 1 < p < q <∞ and. We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on m for m to map LP(M) to U(M), and for the example where there exists ß in R+ such that)(t)= for all in R+, we give conditions which ensure that the maximal function suptaTtf is bounded
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