Parabolic Hermite Lipschitz Spaces: Regularity of Fractional Operators
- PDF / 692,034 Bytes
- 41 Pages / 439.37 x 666.142 pts Page_size
- 39 Downloads / 228 Views
Parabolic Hermite Lipschitz Spaces: Regularity of Fractional Operators Marta De Le´on-Contreras and Jos´e L. Torrea Abstract. We introduce a pointwise definition of Lipschitz (also called H¨ older) spaces adapted to the parabolic Hermite operator H = ∂t − the following spaces Δx + |x|2 on Rn+1 . Also for every α > 0, we define √ by means of the Poisson semigroup of H, PyH = e−y H : H
ΛP α =
√ f : f ∈ L∞ (Rn+1 ) and ∂yk e−y H f L∞ (Rn+1 ) −k+α ≤ Ck y , for k = [α] + 1, y > 0 ,
with the obvious norm. We prove that both spaces do coincide and their norms are equivalent. For the harmonic oscillator, H = −Δx +|x|2 , Stinga and Torrea introduced in 2011 adapted H¨ older classes. Parallel to the parabolic case, we characterize these pointwise H¨ older spaces via the √ L∞ norm of the derivatives of the Poisson and heat semigroups, e−y H and e−τ H , respectively. As important applications of these semigroups characterizations, we get regularity results regarding the boundedness in these adapted Lipschitz spaces of operators related to H and H as fractional (positive and negative) powers, Bessel potentials, Hermite Riesz transforms, and Laplace transform multipliers, in a more direct way. The proofs use in a fundamental way the semigroup definition of the operators considered along the paper. The non-convolution structure of the operators produces an extra difficulty on the arguments. Mathematics Subject Classification. Primary 42C05; Secondary 35K08, 42B35. Keywords. H¨ older spaces, Lipschitz spaces, semigroups, fractional laplacian, Schauder estimates.
Research partially supported by Ministerio de Ciencia e Innovaci´ on de Espa˜ na PGC2018099124-B-I00 (MINECO/FEDER) and EPSRC research Grant EP/S029486/1.
0123456789().: V,-vol
205
Page 2 of 41
M. De León-Contreras and J. L. Torrea
MJOM
1. Introduction Classically, the spaces C δ , 0 < δ < 1, are defined as the set of bounded functions ϕ, such that |ϕ(x + z) − ϕ(x)| ≤ C|z|δ x, z ∈ Rn .
(1.1)
These classes are in between of the space of continuous functions, C 0 , and the one of bounded differentiable functions with bounded continuous derivative, C 1 . These spaces are usually called either Lipschitz or H¨older classes. For δ = 1, the natural space was introduced by Zygmund [23, Chapter II], and it is the set of continuous and bounded functions ϕ, such that: |ϕ(x + z) + ϕ(x − z) − 2ϕ(x)| ≤ C|z|, x, z ∈ Rn . This space is commonly known as the Zygmund’s space and denoted by Λ. It can be shown that if we denote by Lip the space of functions satisfying (1.1) for δ = 1, then C 1 Lip Λ, see [8]. When the exponents are bigger than 1, analogously to the case of the spaces C p , the definition of the classes involves the derivatives of the functions. Different notations can be found in the literature, we shall adopt the following. Given 0 < δ < 1, k ∈ N, C k+δ is the set of functions, such that all the derivatives of order k belong to C δ . In the theory of Fourier series, it was a classical fact to analyze the validity of several theorems for the case of Lipsc
Data Loading...
