### Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

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## PROPIEDADES ANALÍTICAS Y GEOMÉTRICAS DE ESPACIOS DE HERZ EN ESPACIOS DE TIPO HOMOGÉNEO

### Alejandra Perini

#### Universidad Nacional del Comahue, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the Euclidean context of $R^n$ the Herz spaces, initially introduced by Herz [H] in the context of the study of Berstein-type theorems and Lipschitz spaces, was characterized by an equivalent norm by a later work by Johnson [J]. Today, this characterization is used as the definition of Herz spaces may be apt to extend the definition of these spaces to the context of other metrics in $R^n$. Indeed, Ragusa [R] defines the Herz spaces associated with parabolic metrics in $R^n$ and studies their applications in the context of obtaining regularity results of weak solutions for parabolic differential equations in divergence form. In this work we extend the definition of Herz spaces to the abstract context of homogeneous type spaces making use of ChristŽs dyadic families [Ch]. We also address some analytical and geometric problems of these spaces related to the characterization and unconditionality of wavelet bases.

More precisely. If $ D $ is a dyadic family of type Christ, defined on $ (X, d, \ mu) $ space of homogeneous type, we give the following definition:

Let $(X, d, \mu )$ be a homogeneous type space, $ 1

As a first result we establish that the spaces $ K_{p, q} $ are Banach spaces and that the norm does not depend on the choice of cubes $ Q_{n}^{i} $ in the sense that for two of such sequences the norms are equivalents. On the other hand, some of the results that we obtain can be summarized in the following theorem:

Theorem: Let $ (X, d \mu) $ space of homogeneous type $ D $ be a dyadic family and $ H $ an associated Haar system. Then the Haar system truncated at the level of zero resolution turns out to be an unconditional basis for Herz spaces and a characterization of the norms $ K_{p, q} $ of a function $ f $ in terms of its wavelet coefficients is valid.

We also explore the geometric conditions on the space $ (X, d, \mu ) $ for obtain that the democracy in the Herz space $ K_{p, q} $ of the truncated Haar system implies that Herz space it is a Lebesgue space.

[Ch] Christ M., A T(b) theorem with remarks on analytic capacity and the Cauchy integral}, Colloq. Math. 60/61 (2), 601--628 (1990).

[H] Herz, C.S., Lipschitz Spaces and BernsteinŽs Theorem and absolutely convergent Fourier Transform, Journal of Math. and Mechanics, Vol 18, No 4 (1968).

[J] Johnson, R., Lipschitz Spaces - Littlewood Paley Spaces and convoluteurs. Procceding London Math., Soc (3) 29 127-141 (1974).

[R] Ragusa M. A., Parabolic Herz Spaces and their applications. Applied Math. Letters 25 1270-1273 (2012).

Joint work with Dr. Luis Nowak (Universidad Nacional del Comahue) and Prof Daniela Fernandez (Universidad Nacional del Comahue).