1 Introduction
Let
${\mathbb R}^n$
be the n-dimensional Euclidean space, and
$B(x,r)$
denote the open ball in
${\mathbb R}^n$
centered at x of radius
$r>0$
. We consider the Riesz potential of order
$\alpha $
on the half space
${\mathbb H} = \{ x=(x',x_n) \in {\mathbb R}^{n-1} \times {\mathbb R}^1: x_n> 0\}$
defined by
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) =\int_{B(x,x_n)} |x-y|^{\alpha-n} f(y) dy \end{align*} $$
for
$0<\alpha <n$
and
$f\in L^1_{\textrm {loc}}({\mathbb H})$
. For
$f\in L^p_{\textrm {loc}}({\mathbb H})$
with
$1<p<\infty $
, Trudinger type inequalities for Riesz potentials of order
$\alpha $
have been studied in the limiting case
$\alpha p = n$
(see e.g., [Reference Edmunds, Gurka and Opic8–Reference Edmunds and Krbec11, Reference Mizuta, Nakai, Ohno and Shimomura17, Reference Mizuta, Nakai, Ohno and Shimomura18, Reference Serrin28]).
Our first aim in this paper is to establish Trudinger’s exponential integrability for
$I_{{\mathbb H},\alpha } f$
of functions f satisfying the weighted
$L^p$
condition
$$ \begin{align} \int_{\mathbb H} |f(y) y_n^\beta|^{p} dy \le 1, \end{align} $$
when
$\alpha p=n$
and
$\beta < (n+1)/(2p')$
, where
$1/p+1/p'=1$
(see Theorem 3.1). Note that
$\omega (y) = |y_n|^{\beta p}$
is not always Muckenhoupt
$A_p$
weight; more precisely,
$\omega $
is not Muckenhoupt
$A_p$
weight when
$\beta \notin (-1/p, 1/p')$
(see Remarks 2.2 and 3.3). For this purpose, we apply the technique by Hedberg in [Reference Adams and Hedberg1] using the central Hardy–Littlewood maximal function
$M_{{\mathbb H}} f$
defined by
$$ \begin{align*} M_{{\mathbb H}} f(x) = \sup_{\{r> 0: B(x,r)\subset {\mathbb H}\}} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| \, dy, \end{align*} $$
where
$|B(x,r)|$
denotes the Lebesgue measure of
$B(x,r)$
. We show the boundedness of the maximal operator
$M_{{\mathbb H}}$
(Theorem 2.5), as an improvement of [Reference Mizuta and Shimomura23, Theorem 2.1]. We also give a Sobolev type inequality for
$I_{{\mathbb H},\alpha } f$
of functions f satisfying (1.1) when
$\alpha p<n$
and
$\beta < (n+1)/(2p')$
(Theorem 3.2). Compare Theorem 3.2 with [Reference Mizuta and Shimomura23, Theorem 2.2] which is a Sobolev type inequality for the fractional maximal function.
In the previous paper [Reference Mizuta and Shimomura24, Theorem 3.4], we proved a Sobolev type inequality for
$I_{{\mathbb H},,\alpha } f$
of functions f satisfying the weighted Morrey condition
$$ \begin{align} \sup_{r> 0, x\in {\mathbb H}} \frac{r^\sigma}{|B(x,r)|}\int_{{\mathbb H}\cap B(x,r)} \left( |f(y)| y_n^\beta \right)^{p} dy \le 1, \end{align} $$
when
$\alpha p<\sigma < (n+1)/2$
and
$\beta < (n+1)/(2p')$
. We refer to [Reference Morrey25] and [Reference Peetre26] for Morrey spaces, which were introduced to estimate solutions of partial differential equations. See also [Reference Chiarenza and Frasca5, Reference Di Fazio and Ragusa12]. Applying our discussions in Theorem 3.1, we study Trudinger’s exponential integrability for
$I_{{\mathbb H},\alpha } f$
of functions f satisfying (1.2) when
$\alpha p=\sigma \le n$
and
$\beta < (n+1)/(2p')$
(see Theorem 4.1), as an improvement of [Reference Mizuta and Shimomura24, Theorem 3.4].
Further, as an application, we establish Trudinger’s inequality for
$I_{{\mathbb H},\alpha } f$
in the framework of double phase functionals
$$ \begin{align} {\Phi}(x,t) = t^p + (b(x) t)^q, \end{align} $$
where
$1 < p<q$
and
$b(\cdot )$
is non-negative, bounded and Hölder continuous of order
$\theta \in (0,1]$
(see Theorems 5.1 and 5.2). Double phase functionals are studied by Baroni, Colombo, and Mingione [Reference Baroni, Colombo and Mingione2, Reference Baroni, Colombo and Mingione3, Reference Colombo and Mingione6, Reference Colombo and Mingione7] regarding the regularity theory of differential equations. See [Reference Mizuta and Shimomura24, Theorem 4.1] for Sobolev’s inequality of
$I_{{\mathbb H},\alpha } f$
in the framework of (1.3). We refer to [Reference Maeda, Mizuta, Ohno and Shimomura16, Reference Mizuta, Ohno and Shimomura20, Reference Mizuta, Ohno and Shimomura21] for related results. Other double phase problems were studied e.g., in [Reference Byun and Lee4, Reference De Filippis and Mingione13–Reference Hästö and Ok15, Reference Mizuta, Nakai, Ohno and Shimomura19, Reference Mizuta and Shimomura22, Reference Ragusa and Tachikawa27].
Throughout this paper, let C denote various constants independent of the variables in question. The symbol
$g \sim h$
means that
$C^{-1}h\le g\le Ch$
for some constant
$C>0$
.
2 Boundedness of the maximal operator in the half space
For later use, it is convenient to see the following result.
Lemma 2.1 [Reference Mizuta and Shimomura23, Lemma 2.3]
For
$\varepsilon> (n-1)/2$
and
$x\in {\mathbb H}$
, set
$$ \begin{align*} I(x) &= \int_{B(x,x_n)} y_n^{\varepsilon - n} dy. \end{align*} $$
Then there exists a constant
$C>0$
such that
$$ \begin{align*} I(x) &\le C x_n^{\varepsilon}. \end{align*} $$
Remark 2.2 Let
$\beta> (n+1)/(2p')$
. If
$f(y) = |y_n|^{-a} $
, then:
-
(1)
$\displaystyle \int _{B(x,x_n)} |f(y) y_n^\beta |^p dy < \infty $
for
$x\in {\mathbb H}$
when
$(\beta - a)p + n> (n-1)/2$
and -
(2)
$\displaystyle \int _{B(x,x_n)} f(y) dy = \infty $
for
$x\in {\mathbb H}$
when
$- a + n \le (n-1)/2$
.
If
$ (n+1)/2 \le a < \beta + (n+1)/(2p)$
, then both (1) and (2) hold.
For
$f\in L^1_{\textrm {loc}}({\mathbb H})$
, the central Hardy–Littlewood maximal function
$M_{{\mathbb H}}f$
is defined by
$$ \begin{align*} M_{{\mathbb H}}f(x)=\sup_{\{r>0: B(x,r)\subset {\mathbb H}\}} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)|\, dy. \end{align*} $$
The mapping
$f \mapsto M_{{\mathbb H}} f$
is called the fractional central maximal operator.
The usual fractional maximal function
$Mf$
is defined by
$$ \begin{align*} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{{\mathbb H}\cap B(x,r)} |f(y)|\, dy. \end{align*} $$
It is known that the maximal operator
$f \to Mf$
is bounded in Morrey spaces as follows:
Lemma 2.3 [Reference Di Fazio and Ragusa12, Lemma 4]
Let
$0 < \sigma \le n$
and
$p> 1$
. Then there exists a constant
$C> 0$
such that
$$ \begin{align*} \sup_{z\in {\mathbb R}^n, r> 0} \frac{r^\sigma}{|B(z,r)|} \int_{B(z,r)} \{M f(x)\}^p dx &\le C \sup_{z\in {\mathbb R}^n, r > 0} \frac{r^\sigma}{|B(z,r)|} \int_{B(z,r)} |f(y)|^p dy \end{align*} $$
for all measurable functions f on
${\mathbb R}^n$
.
Throughout this paper, let
$1 < p < \infty $
and
$1/p + 1/p' = 1$
. We extend Lemma 2.3 to
$M_{{\mathbb H}}$
. For this purpose, we prepare the following result.
Lemma 2.4 Let
$ \beta < (n+1)/(2p')$
. Then there exists a constant
$C> 0$
such that
$$ \begin{align*} M_{{\mathbb H}} f(x) &\le C x_n^{-\beta} \left( Mg(x) \right)^{1/p} \end{align*} $$
for all
$x\in {\mathbb H}$
and measurable functions f on
${\mathbb H}$
, where
$g (y) = (|f(y)| |y_n|^{\beta })^{p} \chi _{{\mathbb H}}(y)$
.
Proof Let f be a non-negative measurable function on
${\mathbb H}$
. For
$0 < r < x_n/2$
,
$$ \begin{align*} \int_{B(x,r)} y_n^{-\beta p'} dy &\le C x_n^{-\beta p'} r^n \end{align*} $$
and for
$x_n/2 < r < x_n$
and
$-\beta p' + n> (n-1)/2$
$$ \begin{align*} \int_{B(x,r)} y_n^{-\beta p'} dy &\le C x_n^{-\beta p' + n} \le C x_n^{-\beta p'} r^n \end{align*} $$
by Lemma 2.1. Hence, we have by Hölder’s inequality
$$ \begin{align*} \int_{B(x,r)} f(y) dy &\le \left(\int_{B(x,r)} y_n^{-\beta p'} dy \right)^{1/p'} \left(\int_{B(x,r)} (f(y) y_n^{\beta})^{p} dy \right)^{1/p} \\ &\le Cx_n^{-\beta} r^{n/p'} \left(\int_{B(x,r)} (f(y) y_n^{\beta})^{p} dy \right)^{1/p} , \end{align*} $$
so that
$$ \begin{align*} M_{{\mathbb H}} f(x) &\le C x_n^{-\beta} \sup_{0 < r < x_n} \left( \frac{1}{|B(x,r)|} \int_{B(x,r)} (f(y) y_n^{\beta})^{p} dy \right)^{1/p}, \end{align*} $$
as required.▪
By Lemmas 2.3 and 2.4, we obtain the following result, which is an improvement of [Reference Mizuta and Shimomura23, Theorem 2.1].
Theorem 2.5 Let
$ \beta < (n+1)/(2p')$
and
$0 < \sigma \le n$
. Then there exists a constant
$C> 0$
such that
$$ \begin{align*} & \sup_{r>0,z\in {\mathbb H}} \frac{r^\sigma}{|B(z,r)|} \int_{{\mathbb H}\cap B(z,r)} | x_n^\beta M_{{\mathbb H}} f(x) |^{p} dx \\[3pt] & \quad \le C\sup_{r>0,z\in {\mathbb H}} \frac{r^\sigma}{|B(z,r)|}\int_{{\mathbb H}\cap B(z,r)} |f(y) y_n^\beta|^{p} dy \end{align*} $$
for all measurable functions f on
${\mathbb H}$
.
Proof Let f be a measurable function on
${\mathbb H}$
, and take q such that
$1 < q < p$
and
$\beta < (n+1)/(2q')$
. Lemma 2.4 with p replaced by q and Lemma 2.3 give
$$ \begin{align*} & \sup_{r>0,z\in {\mathbb H}} \frac{r^\sigma}{|B(z,r)|} \int_{{\mathbb H}\cap B(z,r)} | x_n^\beta M_{{\mathbb H}} f(x) |^{p} dx \\[3pt] & \quad \le C\sup_{r>0,z\in {\mathbb H}} \frac{r^\sigma}{|B(z,r)|} \int_{{\mathbb H}\cap B(z,r)} |Mg(x)|^{p/q} dx \\[3pt] & \quad \le C \sup_{r>0,z\in {\mathbb H}} \frac{r^\sigma}{|B(z,r)|} \int_{{\mathbb H}\cap B(z,r)} |g(y)|^{p/q} dy \\[3pt] & \quad = C\sup_{r>0,z\in {\mathbb H}} \frac{r^\sigma}{|B(z,r)|}\int_{{\mathbb H}\cap B(z,r)} |f(y) y_n^\beta|^{p} dy, \end{align*} $$
where
$g (y) = ( |f(y)| |y_n|^{\beta })^{q} \chi _{{\mathbb H}}(y)$
.▪
3 Trudinger’s inequality for Riesz potentials in
$L^p$
For
$0<\alpha <n$
and
$f\in L^1_{\textrm {loc}}({\mathbb H})$
, let us consider the Riesz potential of order
$\alpha $
on
${\mathbb H}$
defined by
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) =\int_{B(x,x_n)} |x-y|^{\alpha-n} f(y) dy. \end{align*} $$
We are now ready to show Trudinger’s exponential integrability for Riesz potentials on
${\mathbb H}$
.
Theorem 3.1 Let
$ \alpha p =n$
and
$ \beta < (n+1)/(2p')$
. Then there exist constants
$c_1>0, c_2>0$
such that
$$ \begin{align*} \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left( \left|x_n^{\beta} I_{{\mathbb H},\alpha} f(x)/c_1 \right|{}^{p'} \right) dx &\le c_2 \end{align*} $$
for all
$R> 0$
and measurable functions f satisfying (1.1).
Proof Let
$\alpha p = n$
and f be a non-negative measurable function on
${\mathbb H}$
satisfying (1.1). Write
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) &=\int_{B(x,r)} |x-y|^{\alpha-n} f(y) dy + \int_{B(x,x_n)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) dy \\[3pt] &= T_1(x) + T_2(x). \end{align*} $$
First note that
$$ \begin{align*} T_1(x) &\le C r^\alpha M_{\mathbb H} f(x). \end{align*} $$
Next, we have by Hölder’s inequality for
$0 < r < x_n/2$
$$ \begin{align*} T_{21}(x) &= \int_{B(x,x_n/2)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) dy \\[3pt] &\le C x_n^{-\beta} \int_{B(x,x_n/2)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) y_n^\beta dy \\[3pt] &\le C x_n^{-\beta} \left( \int_{B(x,x_n/2)\setminus B(x,r)} |x-y|^{(\alpha-n)p'} dy \right)^{1/p'} \\[3pt] & \quad \times \left( \int_{B(x,x_n/2)\setminus B(x,r)} \{f(y) y_n^\beta\}^p dy \right)^{1/p} \\[3pt] &\le C x_n^{-\beta} \left( \log (x_n/r) \right)^{1/p'}, \end{align*} $$
since
$\alpha p = n$
. Moreover, we have by Hölder’s inequality and Lemma 2.1 for
$x_n/2 \le r < x_n$
$$ \begin{align*} T_{22}(x) &= \int_{B(x,x_n)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) dy \\[3pt] &\le C x_n^{\alpha-n} \int_{B(x,x_n)\setminus B(x,r)} f(y) dy \\[3pt] &\le C x_n^{\alpha-n} \left( \int_{B(x,x_n)\setminus B(x,r)} y_n^{-\beta p'} dy \right)^{1/p'} \left( \int_{B(x,x_n)\setminus B(x,r)} \{ f(y) y_n^\beta\} ^p dy \right)^{1/p} \\[3pt] &\le C x_n^{\alpha-n} \left( x_n^{-\beta p' + n} \right)^{1/p'} \left( \int_{B(x,x_n)\setminus B(x,r)} \{ f(y) y_n^\beta\} ^p dy \right)^{1/p} \\[3pt] &\le C x_n^{-\beta}, \end{align*} $$
since
$-\beta p' + n> (n-1)/2$
and
$\alpha p = n$
. Therefore,
$$ \begin{align*} T_2(x) &\le C x_n^{-\beta} \left\{ \log (e+(x_n/r)) \right\}^{1/p'} , \end{align*} $$
so that
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) &\le C r^\alpha M_{\mathbb H} f(x) + C x_n^{-\beta} \left\{ \log (e+(x_n/r)) \right\}^{1/p'} \end{align*} $$
for every
$r> 0$
. Letting
$r = \{x_n^{\beta } M_{\mathbb H} f(x)\}^{-1/\alpha }$
, we obtain
$$ \begin{align*} x_n^{\beta} I_{{\mathbb H},\alpha} f(x) &\le C+ C \left(\log (e+x_n \{x_n^{\beta} M_{\mathbb H} f(x)\}^{1/\alpha}) \right)^{1/p'}. \end{align*} $$
Hence, there exists a constant
$c_1>0$
such that
$$ \begin{align*} x_n^{\beta} I_{{\mathbb H},\alpha} f(x) &\le c_1 \left( \log (e+x_n^{\alpha p} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p ) \right)^{1/p'} , \end{align*} $$
so that
$$ \begin{align*} \exp \left[ \left\{x_n^{\beta} I_{{\mathbb H},\alpha} f(x)/c_1 \right\}^{p'} \right] &\le e + x_n^{n} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p, \end{align*} $$
since
$\alpha p = n$
. Now it follows from Theorem 2.5 that
$$ \begin{align*} & \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left[ \left\{x_n^{\beta} I_{{\mathbb H},\alpha} f(x)/c_1 \right\}^{p'} \right] dx \\ &\le e + \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} x_n^{n} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p dx \\ &\le e + C \int_{{\mathbb H}\cap B(0,R)} \{x_n^{\beta} M_{\mathbb H} f(x)\}^{p} dx \\ &\le c_2 \end{align*} $$
for
$R> 0$
.▪
In the same manner as the previous proof, we obtain Sobolev’s inequality in weighted
$L^p$
spaces.
Theorem 3.2 (cf. [Reference Mizuta and Shimomura23, Theorem 2.2])
Let
$1/p^* = 1/p - \alpha /n> 0$
and
$ \beta < (n+1)/ (2p')$
. Then there exists a constant
$C>0$
such that
$$ \begin{align*} \int_{{\mathbb H}} \left| x_n^{\beta} I_{{\mathbb H},\alpha} f(x) \right|{}^{p^*} dx \le C \end{align*} $$
for all measurable functions f satisfying (1.1).
In fact, as in the proof of Theorem 3.1, we have by Hölder’s inequality
$$ \begin{align*} T_{21}(x) \le C x_n^{-\beta} r^{\alpha-n/p} \end{align*} $$
for
$0 < r < x_n/2$
, and
$$ \begin{align*} T_{22}(x) \le C x_n^{-\beta} x_n^{\alpha - n/p} \end{align*} $$
for
$x_n/2 \le r < x_n$
. Hence,
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) &\le C r^\alpha M_{\mathbb H} f(x) + C x_n^{-\beta} r^{\alpha - n/p} \end{align*} $$
for every
$r> 0$
. Letting
$r = \{x_n^{\beta } M_{\mathbb H} f(x)\}^{-p/n}$
, we obtain
$$ \begin{align*} x_n^{\beta} I_{{\mathbb H},\alpha} f(x) &\le C\{x_n^{\beta} M_{\mathbb H} f(x)\}^{1-\alpha p/n} \\ &= C \{x_n^{\beta} M_{\mathbb H} f(x)\}^{p/p^*}. \end{align*} $$
Now it follows from Theorem 2.5 that
$$ \begin{align*} \int_{{\mathbb H}} \left\{ x_n^{\beta} I_{{\mathbb H},\alpha} f(x) \right\}^{p^*} dx &\le C \int_{{\mathbb H}} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p dx \\[2pt] &\le C \int_{{\mathbb H}} |y_n^{\beta} f(y)|^p dy. \end{align*} $$
Remark 3.3 Let
$\beta + \alpha - b + n/q \le a < \beta -b+ n/p$
and
$(n-1)/q < (n-1)/p < b$
. If
$f(y) = |y_n|^{-a} |y|^{-b} \chi _{B(0,1)}(y)$
, where
$\chi _E$
denotes the characteristic function of E, then:
-
(1)
$\displaystyle \int _{{\mathbb H}\cap B(0,1)} |f(y) y_n^\beta |^p dy < \infty $
when
$-bp + (n-1) < 0$
and
$(-a+\beta )p + (-b p + n-1) + 1> 0$
; -
(2)
$\displaystyle I_\alpha f(x) = \int _{\mathbb H} |x-y|^{\alpha -n} f(y) dy = \infty $
for all
$x\in {\mathbb H}$
when
$a \ge 1$
; -
(3)
$\displaystyle I_{{\mathbb H},\alpha } f(x) \ge C x_n^{\alpha -a} |x|^{-b} $
for all
$x\in {\mathbb H}\cap B(0,1)$
; -
(4)
$\displaystyle \int _{{\mathbb H}\cap B(0,1)} \{x_n^\beta I_{{\mathbb H},\alpha } f(x) \}^q dx = \infty $
when
$-bq + (n-1) < 0$
and
$(\beta -a + \alpha ) q + (-b q+ n-1) + 1 \le 0$
.In particular, it happens that when
$$ \begin{align*} \int_{{\mathbb H}\cap B(0,1)} \{x_n^\beta I_{{\mathbb H},\alpha} f(x) \}^q dx = \infty, \end{align*} $$
$q> p^*$
.
For (3), it suffices to see that
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) &\ge \int_{B(x,x_n/2)} |x-y|^{\alpha-n} f(y) dy \\[2pt] &\ge C x_n^{-a}|x|^{-b} \int_{B(x,x_n/2)\cap B(0,1)} |x-y|^{\alpha-n} dy \\[2pt] &\ge C x_n^{-a+\alpha} |x|^{-b}. \end{align*} $$
4 Trudinger’s inequality for Riesz potentials in Morrey spaces
In this section, we are concerned with Trudinger’s exponential integrability in weighted Morrey spaces.
Theorem 4.1 Let
$\alpha p = \sigma \le n$
and
$ \beta < (n+1)/(2p')$
. Then there exist constants
$c_1>0, c_2>0$
such that
$$ \begin{align*} \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left(|x_n^{\beta} I_{{\mathbb H},\alpha} f(x)/c_1| \right) dx &\le c_2 \end{align*} $$
for all
$R> 0$
and measurable functions f on
${\mathbb H}$
satisfying (1.2).
Proof Let f be a non-negative measurable function on
${\mathbb H}$
satisfying (1.2). Write
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) &= \int_{B(x,r)} |x-y|^{\alpha-n} f(y) dy + \int_{B(x,x_n)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) dy \\[2pt] &= T_1(x) + T_2(x). \end{align*} $$
By (1.2), we have for
$0 < r < x_n/2$
$$ \begin{align*} T_{21}(x) &= \int_{B(x,x_n/2)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) dy \\[2pt] &\le C x_n^{-\beta} \int_{B(x,x_n/2)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) y_n^\beta dy \\[2pt] &\le C x_n^{-\beta} \int_r^{x_n/2} t^{\alpha-1} \left( \frac{1}{|B(x,t)|} \int_{B(x,t)} f(y) y_n^\beta dy \right) dt \\[2pt] &\le C x_n^{-\beta} \int_r^{x_n/2} t^{\alpha-1} \left( \frac{1}{|B(x,t)|} \int_{B(x,t)} \{f(y) y_n^\beta\}^p dy \right)^{1/p} dt \\[2pt] &\le C x_n^{-\beta} \int_r^{x_n/2} t^{-1} dt \\[2pt] &\le C x_n^{-\beta} \log (x_n/r), \end{align*} $$
since
$\alpha p = \sigma $
. Moreover, as in the proof of Theorem 3.1, by Hölder’s inequality and Lemma 2.1, we have for
$x_n/2 \le r < x_n$
$$ \begin{align*} T_{22}(x) &= \int_{B(x,x_n)\setminus B(x,r)} |x-y|^{\alpha-n} f(y) dy \\[2pt] &\le C x_n^{\alpha-n} \int_{B(x,x_n)\setminus B(x,r)} f(y) dy \\[2pt] &\le C x_n^{\alpha-n} \left( x_n^{-\beta p' + n} \right)^{1/p'} \left( \int_{B(x,x_n)\setminus B(x,r)} \{ f(y) y_n^\beta\} ^p dy \right)^{1/p} \\[2pt] &\le C x_n^{\alpha-n} \left( x_n^{-\beta p' + n} \right)^{1/p'} (x_n^{n-\sigma})^{1/p} \\[2pt] &= C x_n^{-\beta}, \end{align*} $$
since
$-\beta p' + n> (n-1)/2$
and
$\alpha p = \sigma $
. Therefore,
$$ \begin{align*} T_2(x) &\le C x_n^{-\beta} \log (x_n/r) , \end{align*} $$
so that
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) &\le C r^\alpha M_{\mathbb H} f(x) + C x_n^{-\beta} \log (e +(x_n/r)) \end{align*} $$
for every
$r> 0$
. Letting
$r = \{x_n^{\beta } M_{\mathbb H} f(x)\}^{-1/\alpha }$
, we obtain
$$ \begin{align*} x_n^{\beta} I_{{\mathbb H},\alpha} f(x) &\le C+ C \log (e+x_n \{x_n^{\beta} M_{\mathbb H} f(x)\}^{1/\alpha}). \end{align*} $$
Hence, there exists a constant
$c_1>0$
such that
$$ \begin{align*} x_n^{\beta} I_{{\mathbb H},\alpha} f(x) &\le c_1\log (e+x_n^{\alpha p} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p ), \end{align*} $$
so that
$$ \begin{align*} \exp \left(x_n^{\beta} I_{{\mathbb H},\alpha} f(x)/c_1 \right) &\le e + x_n^{\alpha p} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p. \end{align*} $$
Hence, in view of Theorem 2.5, we obtain
$$ \begin{align*} & \frac{1}{|B(0,R)|} \int_{{\mathbb H}\cap B(0,R)} \exp \left(x_n^{\beta} I_{{\mathbb H},\alpha} f(x)/c_1 \right) dx \\[2pt] &\le e + C \frac{1}{|B(0,R)|} \int_{{\mathbb H}\cap B(0,R)} x_n^{\sigma} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p dx \\[2pt] &\le e + C \frac{R^{\sigma}}{|B(0,R)|} \int_{{\mathbb H}\cap B(0,R)} \{x_n^{\beta} M_{\mathbb H} f(x)\}^{p} dx \\[2pt] &\le c_2 \end{align*} $$
for
$R> 0$
.▪
In the same manner as the previous proof, we obtain Sobolev’s inequality in weighted Morrey spaces, which is an improvement of [Reference Mizuta and Shimomura24, Theorem 3.4].
Theorem 4.2 Let
$1/p_{\sigma } = 1/p - \alpha /\sigma> 0$
,
$0 < \sigma \le n$
and
$ \beta < (n+1)/(2p')$
. Then there exists a constant
$C>0$
such that
$$ \begin{align*} \frac{r^\sigma}{|B(z,r)|}\int_{{\mathbb H}\cap B(z,r)} \left|x_n^{\beta} I_{{\mathbb H},\alpha} f(x) \right|{}^{p_{\sigma}} dx \le C \end{align*} $$
for all
$z\in {\mathbb H}$
,
$r> 0$
and measurable functions f on
${\mathbb H}$
satisfying (1.2).
In fact, as in the proof of Theorem 4.1, we have by Hölder’s inequality
$$ \begin{align*} T_{21}(x) \le C x_n^{-\beta} r^{\alpha-\sigma/p} \end{align*} $$
for
$0 < r < x_n/2$
, and
$$ \begin{align*} T_{22}(x) \le C x_n^{-\beta} x_n^{\alpha - \sigma/p} \end{align*} $$
for
$x_n/2 \le r < x_n$
. Hence,
$$ \begin{align*} I_{{\mathbb H},\alpha} f(x) &\le C r^\alpha M_{\mathbb H} f(x) + C x_n^{-\beta} r^{\alpha - \sigma/p} \end{align*} $$
for every
$r> 0$
. Letting
$r = \{x_n^{\beta } M_{\mathbb H} f(x)\}^{-p/\sigma }$
, we obtain
$$ \begin{align*} x_n^{\beta} I_{{\mathbb H},\alpha} f(x) &\le C\{x_n^{\beta} M_{\mathbb H} f(x)\}^{1-\alpha p/\sigma} \\[2pt] &= C \{x_n^{\beta} M_{\mathbb H} f(x)\}^{p/p_{\sigma}}. \end{align*} $$
Now it follows from Theorem 2.5 that
$$ \begin{align*} \frac{r^\sigma}{|B(z,r)|}\int_{{\mathbb H}\cap B(z,r)} \left\{ x_n^{\beta} I_{{\mathbb H},\alpha} f(x) \right\}^{p_{\sigma}} dx &\le C \frac{r^\sigma}{|B(z,r)|}\int_{{\mathbb H}\cap B(z,r)} \{x_n^{\beta} M_{\mathbb H} f(x)\}^p dx \\[2pt] &\le C \frac{r^\sigma}{|B(z,r)|}\int_{{\mathbb H}\cap B(z,r)} |y_n^{\beta} f(y)|^p dy \end{align*} $$
for all
$z\in {\mathbb H}$
and
$r> 0$
.
5 Double phase functionals
In this section, we consider the double phase functional
$$ \begin{align*} \Phi(x,t) = t^{p} + (b(x) t)^{q}, \end{align*} $$
where
$1 < p<q$
and
$b(\cdot )$
is non-negative, bounded and Hölder continuous of order
$\theta \in (0,1]$
[Reference Maeda, Mizuta, Ohno and Shimomura16].
We obtain Trudinger’s inequality for
$I_{{\mathbb H},\alpha } f$
in weighted Morrey spaces of the double phase functional
$\Phi (x,t)$
using Theorem 4.1.
Theorem 5.1 Let
$0 < \sigma \le n$
,
$1/q = 1/p - \theta /\sigma $
,
$1/p_\sigma = 1/p - \alpha /\sigma> 0$
and
$1/q_\sigma = 1/q - \alpha /\sigma = 0$
. Suppose
$ \beta < (n+1)/(2p')$
. Then there exist constants
$c_1>0, c_2>0$
such that
$$ \begin{align*} & \frac{R^\sigma}{|B(0,R)|} \int_{{\mathbb H}\cap B(0,R)} \left|x_n^{\beta} I_{{\mathbb H},\alpha} f(x) \right|{}^{p_\sigma} dx \\[2pt] & \quad + \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left( |x_n^\beta b(x) I_{{\mathbb H},\alpha} f(x)/c_1| \right) dx \le c_2 \end{align*} $$
for all
$R>0$
and measurable functions f on
${\mathbb H}$
satisfying
$$ \begin{align} \sup_{x\in {\mathbb H},r>0} \frac{r^\sigma}{|B(x,r)|}\int_{{\mathbb H}\cap B(x,r)} \Phi\left(y, |f(y)| y_n^\beta \right) dy \le 1. \end{align} $$
Proof Let f be a non-negative measurable function on
${\mathbb H}$
satisfying (5.1). First, we see from Theorem 4.2 that
$$\begin{align*}\sup_{r>0: x\in {\mathbb H}} \frac{r^\sigma}{|B(x,r)|}\int_{{\mathbb H}\cap B(x,r)} (z_n^\beta I_{{\mathbb H},\alpha} f(z))^{p_\sigma} dz \le C, \end{align*}$$
since
$\alpha p<\sigma $
.
Note that
$$ \begin{align*} & b(x) I_{{\mathbb H},\alpha} f(x) \\[2pt] & \quad = \int_{B(x,x_n)} \{b(x)-b(y)\} |x-y|^{\alpha-n} f(y) dy + \int_{B(x,x_n)} b(y)|x-y|^{\alpha-n}f(y) dy \\[2pt] & \quad \le C \int_{B(x,x_n)} |x-y|^{\alpha+\theta-n} f(y) dy + \int_{B(x,x_n)} |x-y|^{\alpha-n} b(y)f(y) dy \\[2pt] & \quad = C I_{{\mathbb H},\alpha+\theta} f(x)+I_{{\mathbb H},\alpha} [bf](x), \end{align*} $$
when
$x\in {\mathbb H}$
. We find by Theorem 4.1
$$ \begin{align*} \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left(|x_n^{\beta} I_{{\mathbb H},\alpha+\theta} f(x)/c_1| \right) dx &\le c_2 \end{align*} $$
and
$$ \begin{align*} \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left(|x_n^{\beta} I_{{\mathbb H},\alpha} [bf](x)/c_1| \right) dx &\le c_2 \end{align*} $$
for all
$R>0$
since
$1/q_\sigma = 1/q - \alpha /\sigma = 1/p - (\alpha +\theta )/\sigma = 0$
. Thus, we complete the proof.▪
We obtain the following theorem using Theorem 3.1.
Theorem 5.2 Let
$1/q = 1/p - \theta /n$
,
$ \alpha p <n=\alpha q$
and
$ \beta < (n+1)/(2p')$
. Then there exist constants
$c_1>0, c_2>0$
such that
$$ \begin{align*} & \int_{{\mathbb H}} \left|x_n^{\beta} I_{{\mathbb H},\alpha} f(x) \right|{}^{p^*} dx \\ & {} + \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left( \left|x_n^{\beta} b(x) I_{{\mathbb H},\alpha} f(x)/c_1 \right|{}^{q'} \right) dx \le c_2 \end{align*} $$
for all
$R>0$
and measurable functions f satisfying
$$ \begin{align} \int_{{\mathbb H}} \Phi\left(y, |f(y)| y_n^\beta \right) dy \le 1. \end{align} $$
Proof Let f be a non-negative measurable function on
${\mathbb H}$
satisfying (5.2). Then Theorem 3.2 gives
$$ \begin{align*} \int_{{\mathbb H}} \left|x_n^{\beta} I_{{\mathbb H},\alpha} f(x) \right|{}^{p^*} dx \le C, \end{align*} $$
since
$\alpha p<n$
. Further, Theorem 3.1 gives
$$ \begin{align*} \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left( \left|x_n^{\beta} I_{{\mathbb H},\alpha+\theta} f(x)/c_1 \right|{}^{p'} \right) dx &\le c_2 \end{align*} $$
and
$$ \begin{align*} \frac{1}{|B(0,R)|}\int_{{\mathbb H}\cap B(0,R)} \exp \left( \left|x_n^{\beta} I_{{\mathbb H},\alpha} [bf](x)/c_1 \right|{}^{q'} \right) dx &\le c_2 \end{align*} $$
for all
$R>0$
since
$ (\alpha +\theta ) p=n=\alpha q$
and
$1/p' < 1/q'$
. Thus, we obtain the result.▪
