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International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 27, 2015 12:20–13:00, Ïëåíàðíûå äîêëàäû, Moscow, Steklov Mathematical Institute of RAS

$B_w^u$-function spaces and their interpolation

T. Sobukawa

Waseda University, Japan
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Abstract: Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space. We denote by $Q_r$ the open cube centered at the origin and sidelength $2r$, or the open ball centered at the origin and of radius $r$, that is,
\begin{align*} Q_r &=\{y=(y_1,y_2,…,y_n)\in\mathbb{R}^n:\max_{1\le i\le n} |y_i|<r \}\qquador
Q_r&=\{y\in\mathbb{R}^n:|y|<r\}. \end{align*}

For each $r\in(0,\infty)$, let $E(Q_r)$ be a function space on $Q_r$ with quasi-norm $\|\cdot\|_{E(Q_r)}$. Let $E_{Q}(\mathbb{R}^n)$ be the set of all measurable functions $f$ on $\mathbb{R}^n$ such that $f|_{Q_r}\in E(Q_r)$ for all $r>0$.
We assume the following restriction property:
\label{N384:restriction} f|_{Q_r}\in E(Q_r) and 0<t<r<\infty\quad \Longrightarrow\quad f|_{Q_t}\in E(Q_t) and \|f\|_{E(Q_t)}\le C_E\|f\|_{E(Q_r)},
where $C_E$ is a positive constant independent of $r$, $t$ and $f$.
Definition. Let $w\colon (0,\infty)\to(0,\infty)$ be a weight function and let $u\in(0,\infty]$. We define function spaces ${B_{w}^{u}(E)}=B_{w}^{u}(E)(\mathbb{R}^n)$ and $\dot B_w^u=\dot{B}_{w}^{u}(E)(\mathbb{R}^n)$ as the sets of all functions $f\in E_Q(\mathbb{R}^n)$ such that $\|f\|_{B_{w}^{u}(E)}<\infty$ and $\|f\|_{\dot{B}_{w}^{u}(E)}<\infty$, respectively, where
\begin{align*} \|f\|_{B_{w}^{u}(E)}&=\|w(r)\|f\|_{E(Q_r)}\|_{L^{u}([1,\infty),dr/r)},
\|f\|_{\dot{B}_{w}^{u}(E)}&=\|w(r)\|f\|_{E(Q_r)}\|_{L^{u}((0,\infty),dr/r)}. \end{align*}
In the above we abbreviated $\|f|_{Q_r}\|_{E(Q_r)}$ to $\|f\|_{E(Q_r)}$.
If $E=L^p$, then $\dot{B}_{w}^{u}(L^p)(\mathbb{R}^n)$ is the local Morrey-type space introduced by Burenkov and Guliyev [6], Example 5, below.
Here, we always assume that $w$ has some decreasing condition. Note that, if $w(r)\to\infty$ as $r\to\infty$, then ${B_{w}^{u}(E)}=\dot{B}wu=\{0\}$.
In particular, if $w(r)=r^{-\sigma}$, $\sigma\ge0$ and $u=\infty$, we denote $B_{w}^{u}(E)(\mathbb{R}^n)$ and $\dot{B}_{w}^{u}(E)(\mathbb{R}^n)$ by $B_{\sigma}(E)(\mathbb{R}^n)$ and $\dot{B}_{\sigma}(E)(\mathbb{R}^n)$, respectively, which were introduced recently by Komori-Furuya, Matsuoka, Nakai and Sawano [17]. These $B_{\sigma}$-function spaces unify several function spaces, see the following Examples 1–4.
Example 1. $B^p(\mathbb{R}^n)$, the dual of Beuling algebra $A^p(\mathbb{R}^n)$ (Beurling [2], Feichtinger [12]).
Example 2. The central mean oscillation space $\mathrm{CMO}^{p}(\mathbb{R}^n)$, the central bounded mean oscillation space $\mathrm{CBMO}^{p}(\mathbb{R}^n)$ (Chen and Lau [11] and García-Cuerva [13] , Lu and Yang [19], [20]).
Example 3. The central Morrey spaces, the $\lambda$-central mean oscillation space and the $\lambda$-central bounded mean oscillation space as an extension of $B^p(\mathbb{R}^n)$, $\dot{B}^p(\mathbb{R}^n)$, $\mathrm{CMO}^p(\mathbb{R}^n)$ and $\mathrm{CBMO}^p(\mathbb{R}^n)$ (García-Cuerva and Herrero [14] and Alvarez, Guzmán-Partida and Lakey [1]).
Example 4. If $E=L_{p,\lambda}$ (Morrey space) or $\mathcal{L}_{p,\lambda}$ (Campanato space), then the function spaces $B_{\sigma}(L_{p,\lambda})(\mathbb{R}^n)$, $\dot{B}_{\sigma}(L_{p,\lambda})(\mathbb{R}^n)$, $B_{\sigma}(\mathcal{L}_{p,\lambda})(\mathbb{R}^n)$ and $\dot{B}_{\sigma}(\mathcal{L}_{p,\lambda})(\mathbb{R}^n)$ unify the function spaces in above examples and the usual Morrey-Campanato and Lipschitz spaces. Actually, if $\lambda=-n/p$, then $L_{p,\lambda}=L^p$. If $\sigma=0$, then
\begin{align*} B_{0}(L_{p,\lambda})(\mathbb{R}^n)&=\dot{B}_{0}(L_{p,\lambda})(\mathbb{R}^n) =L_{p,\lambda}(\mathbb{R}^n),
B_{0}(\mathcal{L}_{p,\lambda})(\mathbb{R}^n)&=\dot{B}_{0}(\mathcal{L}_{p,\lambda})(\mathbb{R}^n) =\mathcal{L}_{p,\lambda}(\mathbb{R}^n). \end{align*}
If $\lambda=0$, then $\mathcal{L}_{p,\lambda}(\mathbb{R}^n)=\mathrm{BMO}(\mathbb{R}^n)$ for all $p\in[1,\infty)$ (John and Nirenberg [N384:JohnNirenberg1961]). If $\lambda=\alpha\in(0,1]$, then $\mathcal{L}_{p,\lambda}(\mathbb{R}^n)=\mathrm{Lip}_{\alpha}(\mathbb{R}^n)$ for all $p\in[1,\infty)$ (Campanato [10], Meyers [22], Spanne [24]). $B_{\sigma}$-Morrey-Campanato spaces were investigated in [16], [17], [18], [21].
Example 5. Local Morrey-type space $LM_{p\theta,w}(\mathbb{R}^n)$ with the (quasi-)norm
\begin{equation*} \|f\|_{LM_{p\theta,w}} = \|w(r)\|f\|_{L^p(Q_r)}\|_{L^{\theta}(0,\infty)}, \end{equation*}
(Burenkov and Guliyev [6]). $LM_{p\theta,\widetilde{w}}(\mathbb{R}^n)$ is expressed by $\dot{B}_{w}^{u}(E)(\mathbb{R}^n)$ with $E=L^p$ and $\widetilde{w}(r)=w(r)/r$. For recent progress of local Morrey-type spaces, see [3], [4]. See also [5], [9] for interpolation spaces for local Morrey-type spaces.
In this talk, we treat the interpolation property of $B_w^u$-function spaces. To do this, we also the following decomposition property: For any $f\in E_Q(\mathbb{R}^n)$ and for any $r>0$, there exists a decomposition $f=f_0^r+f_1^r$ such that
\label{N384:decomposition0} \|f_0^r\|_{E(Q_t)}\le \begin{cases} C_E\|f\|_{E(Q_t)} & (0<t<r),
C_E\|f\|_{E(Q_{ar})} & (r\le t<\infty), \end{cases}
and
\label{N384:decomposition1} \|f_1^r\|_{E(Q_t)}\le \begin{cases} 0 & (0<t<cr),
C_E\|f\|_{E(Q_{bt})} & (cr\le t<\infty), \end{cases}
where $C_E$, $a$, $b$, $c$ are positive constants independent of $r$, $t$ and $f$.
Theorem. Assume that a family $\{(E(Q_r),\|\cdot\|_{E(Q_r)})\}_{0<r<\infty}$ has the restriction and decomposition properties above. Let $u_0,u_1,u\in(0,\infty]$, $w_0,w_1\in\mathcal{W}^{\infty}$, and
\begin{equation*} w=w_0^{1-\theta}w_1^\theta. \end{equation*}
Assume also that, for some positive constant $\epsilon$, $(w_0(r)/w_1(r))r^{-\epsilon}$ is almost increasing, or, $(w_1(r)/w_0(r))r^{-\epsilon}$ is almost increasing. Then
\begin{equation*} (\dot{B}_{w_0}^{u_0}(E)(\mathbb{R}^n),\dot{B}_{w_1}^{u_1}(E)(\mathbb{R}^n))_{\theta, u} = \dot{B}_{w}^{u}(E)(\mathbb{R}^n), \end{equation*}
and
\begin{equation*} (B_{w_0}^{u_0}(E)(\mathbb{R}^n),B_{w_1}^{u_1}(E)(\mathbb{R}^n))_{\theta, u, [1,\infty)} = B_{w}^{u}(E)(\mathbb{R}^n). \end{equation*}

Here, $(A_0,A_1)_{\theta, u}$ is the usual $K$-real interpolation space, and we define the quasi-norm of $(A_0,A_1)_{u, [1,\infty)}$ as
\begin{equation*} \|a\|_{(A_0,A_1)_{u, [1,\infty)}} =[\int_1^\infty(\dfrac{K(t,a;A_0,A_1)}{t^\theta})\dfrac{dt}{t}]^{1/u} \end{equation*}

As applications of the interpolation property, we also give the boundedness of linear and sublinear operators. It is known that the Hardy–Littlewood maximal operator, fractional maximal operators, singular and fractional integral operators are bounded on $B_{\sigma}$-Morrey–Campanato spaces, see [16], [17], [18], [21]. Interpolate these function spaces, the we get the boundedness of these operators on $B_w^u(L_{p,\lambda})$,$\dot{B}_w^u(L_{p,\lambda})$, $B_w^u(\mathcal{L}_{p,\lambda})$ and $\dot{B}_w^u(\mathcal{L}_{p,\lambda})$, which are also generalization of the results on the local Morrey-type spaces $LM_{pu,w}(\mathbb{R}^n)$.

Materials: abstract.pdf (191.6 Kb)

Language: English

References
1. J. Alvarez, M. Guzmán-Partida and J. Lakey, “Spaces of bounded $\lambda$-central mean oscillation, Morrey spaces, and $\lambda$-central Carleson measures”, Collect. Math., 51 (2000), 1–47
2. A. Beurling, “Construction and analysis of some convolution algebra”, Ann. Inst. Fourier, 14 (1964), 1–32
3. V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I”, Eurasian Math. J., 3:3 (2012), 11–32
4. V. I. Burenkov, “Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. II”, Eurasian Math. J., 4:1 (2013), 21–45
5. V. I. Burenkov, D. K. Darbayeva, E. D. Nursultanov, “Description of interpolation spaces for general local Morrey-type spaces”, Eurasian Math. J., 4:1 (2013), 46–53
6. V. I. Burenkov, H. V. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces”, Studia Math., 163:2 (2004), 157–176
7. V. I. Burenkov, V. S. Guliyev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces”, Potential Anal., 30:3 (2009), 211–249
8. V. I. Burenkov, V. S. Guliyev, A. Serbetci, T. V. Tararykova, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces”, Eurasian Math. J., 1:1 (2010), 32–53
9. V. I. Burenkov, E. D. Nursultanov, “Description of interpolation spaces for local Morrey-type spaces”, Proc. Steklov Inst. Math., 269:1 (2010), 46–56
10. S. Campanato, “Propriet`a di hölderianità di alcune classi di funzioni”, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 175–188 (in Italian)
11. Y. Chen, K. Lau, “Some new classes of Hardy spaces”, J. Func. Anal., 84 (1989), 255–278
12. H. Feichtinger, “An elementary approach to Wiener's third Tauberian theorem on Euclidean $n$-space”, Proceedings of Conference at Cortona 1984, Symposia Mathematica, 29, Academic Press, New York, 1987, 267–301
13. J. García-Cuerva, “Hardy spaces and Beurling algebras”, J. London Math. Soc. (2), 39 (1989), 499–513
14. J. García-Cuerva, M.J. L. Herrero, “A theory of Hardy spaces assosiated to the Herz spaces”, Proc. London Math. Soc., 69 (1994), 605–628
15. F. John, L. Nirenberg, “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426
16. Y. Komori-Furuya, K. Matsuoka, “Strong and weak estimates for fractional integral operators on some Herz-type function spaces”, Proceedings of the Maratea Conference FAAT 2009, Rendiconti del Circolo Mathematico di Palermo, Serie II, Suppl., 82, 2010, 375–385
17. Y. Komori-Furuya, K. Matsuoka, E. Nakai and Y. Sawano,, “Integral operators on $B_{\sigma}$-Morrey–Campanato spaces”, Rev. Mat. Complut., 26:1 (2013), 1–32
18. Y. Komori-Furuya, K. Matsuoka, E. Nakai, Y. Sawano, “Littlewood–Paley theory for $B_\sigma$ spaces”, J. Funct. Spaces Appl., 2013, 859402
19. S. Lu, D. Yang, “The Littlewood–Paley function and $\phi$-transform characterizations of a new Hardy space $HK_2$ associated with the Herz space”, Studia Math., 101:3 (1992), 285–298
20. S. Lu and D. Yang,, “The central BMO spaces and Littlewood-Paley operators”, Approx. Theory Appl., 11 (1995), 72–94
21. K. Matsuoka, E. Nakai, “Fractional integral operators on $B^{p,\lambda}$ with Morrey-Campanato norms,”, Function Spaces IX, Banach Center Publ., 92, Polish Acad. Sci. Inst. Math., Warsaw, 2011, 249–264
22. N. G. Meyers, “Mean oscillation over cubes and Hölder continuity”, Proc. Amer. Math. Soc., 15 (1964), 717–721
23. E. Nakai, T. Sobukawa, $B_w^u$-function spaces and their interpolation, 2014, arXiv: 1410.6327
24. S. Spanne, “Some function spaces defined using the mean oscillation over cubes”, Ann. Scuola Norm. Sup. Pisa (3), 19 (1965), 593–608

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