Maximal estimates for Bilinear Bochner-Riesz means

The convergence of the partial sum operators corresponding to two fold product of $n-$ dimensional Fourier series

to $f(x)g(x)$ as $R\rightarrow\infty$ is a classical problem in Fourier analysis. Here $f$ and $g$ are $1-$periodic functions with respect to each variable belonging to Lebesgue spaces $L^{p_{1}}([0,1]^{n})$ and $L^{p_{2}}([0,1]^{n})$ respectively, where $1\leq p_{1},p_{2}\leq\infty$.

This motivates us to consider the following more general partial sum operators

The partial sum operators are referred to as the bilinear Bochner-Riesz means of order $\alpha$.

The transference principle for bilinear multipliers establishes that the study of $L^{p}$ boundedness of the bilinear Bochner-Riesz means is equivalent to studying the corresponding estimates for the bilinear Bochner-Riesz operator of index $\alpha\geq 0$ defined on ${\mathbb{R}}^{n}$ by

where $f,g\in\mathcal{S}({\mathbb{R}}^{n})$ and $\hat{f}$ denotes the Fourier transform of $f$ given by $\hat{f}(\xi)=\int_{{\mathbb{R}}^{n}}f(x)e^{-2\pi ix.\xi}dx$. When $R=1$, we simply use the notation $\mathcal{B}^{\alpha}$.

In this paper we are interested in investigating the a.e. convergence of $\mathcal{B}^{\alpha}_{R}(f,g)$ as $R\rightarrow\infty$ for $(f,g)$ in $L^{p_{1}}({\mathbb{R}}^{n})\times L^{p_{2}}({\mathbb{R}}^{n})$ for $1\leq p_{1},p_{2}\leq\infty$. In view of the general principle due to Stein we know that in order to study the a.e. convergence of $\mathcal{B}^{\alpha}_{R}(f,g)$ it is enough to prove suitable $L^{p}$ estimates for the corresponding maximal function

More precisely, for $f,g\in\mathcal{S}({\mathbb{R}}^{n})$ we are concerned with the following estimates

Also, the corresponding estimate for the bilinear Bochner-Riesz mean

Here we shall always assume that the exponents $p_{1},p_{2},p$ satisfy $1\leq p_{1},p_{2}\leq\infty$ and $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$. The notation $A\lesssim B$ in the estimates above means that there is an implicit constant $C>0$ such that $A\leq CB.$ The constant $C$ may depend on $\alpha,n,p_{1}$ and $p_{2}.$ We will not keep track of such constants appearing in various inequalities unless we need them in order for the proof to work. In some inequalities in the later part of paper we shall also use the notation $A\lesssim_{a,b}B$ to emphasize the dependence of the implicit constant on the parameters $a$ and $b$.

If the estimate (2) (or (3)) holds we say that the operator $\mathcal{B}^{\alpha}_{*}$ (or $\mathcal{B}^{\alpha}$) is bounded from $L^{p_{1}}({\mathbb{R}}^{n})\times L^{p_{2}}({\mathbb{R}}^{n})\rightarrow L^{p}({\mathbb{R}}^{n})$. Observe that the standard dilation arguments tell us that $\mathcal{B}^{\alpha}:L^{p_{1}}({\mathbb{R}}^{n})\times L^{p_{2}}({\mathbb{R}}^{n})\rightarrow L^{p}({\mathbb{R}}^{n})$ if, and only if $\mathcal{B}^{\alpha}_{R}:L^{p_{1}}({\mathbb{R}}^{n})\times L^{p_{2}}({\mathbb{R}}^{n})\rightarrow L^{p}({\mathbb{R}}^{n})$ where $R>0$ and $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}$.

The question of $L^{p_{1}}({\mathbb{R}}^{n})\times L^{p_{2}}({\mathbb{R}}^{n})\rightarrow L^{p}({\mathbb{R}}^{n})$ boundedness for the operators $\mathcal{B}^{\alpha}$ and $\mathcal{B}^{\alpha}_{*}$ is referred to as the bilinear Bochner-Riesz probem in general. This is one of the outstanding open problems in the theory of bilinear Fourier multipliers. In this paper we make significant progress on the bilinear Bochner-Riesz problem for $\mathcal{B}^{\alpha}_{*}$. We establish new and sharp results for the maximal function $\mathcal{B}^{\alpha}_{*}$.

Note that the operator $\mathcal{B}^{\alpha}_{R}$ takes the following form in the spatial coordinates.

where

Here $J_{\alpha+n}$ denotes the standard Bessel function of order $\alpha+n$.

Using the asymptotic estimates of Bessel functions it is easily verified that for $\alpha>n-\frac{1}{2}$, the kernel $K^{\alpha}_{R}$ is an integrable function with a uniform constant with respect to $R$. Consequently, the estimates (2) and (3) follow for all $1<p_{1},p_{2}\leq\infty$ and $\alpha>n-\frac{1}{2}$, see [1, 13] for detail. The index $\alpha=n-\frac{1}{2}$ is referred to as the critical index for the bilinear Bochner-Riesz problem. Therefore, the bilinear Bochner-Riesz problem concerning estimates (2) and (3) needs to be investigated when $0\leq\alpha\leq n-\frac{1}{2}$.

When $\alpha=0$, the operator $\mathcal{B}^{\alpha}$ is denoted by $\mathcal{B}$ and it is referred to as the bilinear ball multiplier operator. In [8] Grafakos and Li proved the estimate (3) in dimension $n=1$ when $\alpha=0$ and $2\leq p_{1},p_{2},p^{\prime}<\infty$ satisfying the Hölder relation $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p}.$ Here $p^{\prime}$ denotes the conjugate index to $p$ given by $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. As mentioned previously in this paper we shall always assume that the exponents $p_{1},p_{2}$ and $p$ satisfy the Hölder relation unless specified otherwise. We will denote such a triplet by $(p_{1},p_{2},p)$. In [8] authors also proved that $\mathcal{B}$ fails to satisfy (3) for triplets $(2,2,1),(2,\infty,2)$ and $(\infty,2,2)$ when $\alpha=0$. We do not know of any positive result for the bilinear ball multiplier in dimension $n=1$ for exponents lying outside the range mentioned as above. Later, Diestel and Grafakos [4] proved a negative result for boundedness of the operator $\mathcal{B}$ when $n\geq 2$. They exploited Fefferman’s argument for the ball multiplier problem [6] to show that in dimension $n\geq 2$ the estimate (3) cannot hold true for $\alpha=0$ if exactly one of $p_{1},p_{2},p^{\prime}$ is less than $2.$

The study of $L^{p}$ boundedness properties of the operator $\mathcal{B}^{\alpha}$ for $\alpha>0$ was initiated in [1] and they proved several positive and negative results for the boundedness of the bilinear Bochner-Riesz means. In particular, they obtained almost complete result in dimension $n=1$ proving that (3) holds for $\alpha>0$ for all $1\leq p_{1},p_{2},p\leq\infty$, see Theorem 4.1 in [1] for detail. This range of exponents is referred to as the Banach triangle in the theory of bilinear multipliers. Observe that the exponent $p$ has its natural range as $1/2\leq p\leq\infty$ when we allow $p_{1}$ and $p_{2}$ in the range $1\leq p_{1},p_{2}\leq\infty.$ Later, in [18] authors extended these results, specifically in the non-Banach traingle (i.e. when $p<1$) to an improved range of $p_{1},p_{2},p$ and $\alpha$.

Recently, Jeong, Lee, and Vargas [12] further improved the range of exponents when $p_{1},p_{2}\geq 2$ for the estimate (3). They introduced a new approach to address the bilinear Bochner-Riesz problem in dimension $n\geq 2$. Their approach relies on a new decompostion of the bilinear Bochner-Riesz operator into a product of square functions in the pointwise sense. This allows them to obtain new results for the estimate (3) when $n\geq 2$ and $p_{1},p_{2}\geq 2$, see [Section 3, [12]] for detail. In particular, in [12] authors obtained new results by improving the lower bounds on the index $\alpha$. They proved optimal result for the estimate (3) when $p_{1}=p_{2}=2$ and $\alpha>0$ for all $n\geq 2$, see Corollary 1.3 in  [12] (also see Proposition 4.10 in [1]). However, the results in other cases are not optimal. We would like to remark here that their method requires the condition that $p_{1},p_{2}\geq 2$. We will see that with our approach we can recover these results. Moreover, our method allows us to obtain $L^{p}$ estimates for $\mathcal{B}^{\alpha}$ when the exponents are less than $2$. In particular, the estimate (3) for exponents in non-Banach triangle are new in dimension $n=1$, see Theorem 2.2. However, for $n\geq 2$ we do not get improved results for the estimate (3) when $p_{1},p_{2}<2$ as compared to already known results in [1, 18].

Now, coming back to the main theme of the paper let us discuss the known results for the bilinear Bochner-Riesz maximal function concerning the estimate (2). The study of $L^{p}$ boundedness properties of $\mathcal{B}^{\alpha}_{*}$ has been initiated in [9, 10]. In [9] authors proved boundedness of maximal bilinear multiplier operator with a certain decay condition on the multiplier symbol. As an application of their result they proved estimate (2) for $p_{1}=p_{2}=2$ and $\alpha>\frac{2n+3}{4}$. Note that this result is interesting only if $\frac{2n+3}{4}$ is smaller or equal to the critical index $\alpha=n-\frac{1}{2}$. Very recently, Jeong and Lee [11] have obtained improved results in this direction generalizing the previously known results significantly. In order to prove $L^{p}$ estimates in [11] authors have exploited the decomposition of the bilinear Bochner-Riesz operators into square functions as carried out in [12]. In particular, they proved that the estimate (2) holds for $\alpha>1$ for $p_{1}=p_{2}=2$. Note that the condition $\alpha>1$ is far from being sharp.

In this paper our main goal is to improve the range of $p_{1},p_{2},p$ and $\alpha$ in the maximal estimate (2). We establish a new decomposition of the bilinear Bochner-Riesz operators in terms of product of square functions which are closely related with the classical Bochner-Riesz square functions (see section 3 for precise detail). As a result, for each piece of the decomposition parametrised by $j$, the corresponding bilinear operator can be written as superposition of product of localised linear operators $B_{j,\beta}^{R,t}f$ and $B_{t}^{\delta}g$. For more details please refer to (10). The operator $B_{j,\beta}^{R,t}$ turns out to be a slight perturbation of the localised Bochner Riesz operator. A further careful analysis allows us to reduce the $L^{p}$ boundedness of the corresponding square function to the known case of the localised square function of Bochner Riesz operator as defined in [15]. This allows us to improve the result of E. Jeong and S. Lee [11] for the boundedness of maximal Bilinear Bochner Riesz function. See (18) and equation $4.2$ in Proposition $4.1$ in [11] for comparision.

This approach allows us to establish new and sharp results for the estimates (2) and (3) with an improved range of exponents $p_{1},p_{2},p$ and lower bound on $\alpha$. Moreover, it is valid uniformly in all dimensions $n\geq 1$. In particular, we obtain optimal results for $p_{1}=p_{2}=2$ when $n\geq 1$, see Theorems 2.1 and 2.2. We remark that the maximal estimate  (2) in dimension $n=1$ is not known. Our approach allows us to address this case and we obtain new and sharp results for the maximal estimate (2) in dimension $n=1$, see Theorem 2.2.

We first set some notation that are required in order to describe our results. For $1\leq p\leq\infty$ and $n\geq 1$ denote $\alpha(p)=\max\left\{n|\frac{1}{p}-\frac{1}{2}|-\frac{1}{2},0\right\}$. Note that for $n=1$ we have $\alpha(p)=0$ for all $1\leq p\leq\infty$. For $n\geq 2,$ define $p_{0}(n)=2+\frac{12}{4n-6-k}$ where $n\equiv k~{}\text{mod}~{}3,k=0,1,2$. Denote $\mathfrak{p}_{n}=\text{min}~{}\left\{p_{0}(n),\frac{2(n+2)}{n}\right\}.$ For $n\geq 1$ and $1\leq p_{1},p_{2}\leq\infty$ we define

The following theorems concerning the $L^{p}$ boundedness of the bilinear Bochner-Riesz maximal function $\mathcal{B}_{*}^{\alpha}$ are the main results of this paper. We state the results for $n=1$ and $n\geq 2$ separately.

Let $N\geq 20n$ and $I$ be a compact interval of ${\mathbb{R}}$. Define $C^{N}(I):=\{\psi\in C^{\infty}_{c}(I):\sup_{x\in I,0\leq j\leq N}|\frac{d^{j}}{dx^{j}}\psi(x)|\leq 1\}$. Consider the partition of unity for the interval $[0,1],$ i.e., for $t\in[0,1]$ consider

where $\psi\in C^{\infty}_{0}[\frac{1}{2},2]$ and $\psi_{0}\in C_{0}^{\infty}[-\frac{3}{4},\frac{3}{4}].$

Let $D=\sup_{x\in[\frac{1}{2},2],0\leq j\leq N}|\frac{d^{j}}{dx^{j}}\psi(x)|$. Then $D^{-1}\psi\in C^{N}([\frac{1}{2},2])$.

Setting $t=\frac{|\xi|^{2}}{R^{2}}$ in the above we get that

where

and

With the decomposition of the symbols as above we can write

where

First, we deal with the bilinear operator $\mathcal{B}_{1,R}^{\alpha}(f,g)$. We further decompose the multiplier as

where

Therefore we have

where $T^{\alpha}_{j,R}(f,g)(x)=\int_{{\mathbb{R}}^{n}}\int_{{\mathbb{R}}^{n}}\tilde{m}^{\alpha}_{j,R}(\xi,\eta)\hat{f}(\xi)\hat{g}(\eta)e^{2\pi ix.(\xi+\eta)}d\xi d\eta$.

Let us denote $\varphi_{R}(\xi)=\left(1-\frac{|\xi|^{2}}{R^{2}}\right)_{+}.$ Now using an identity from Stein and Weiss [[21], page 278] we have the following relation.

where $\beta>\frac{1}{2}$, $\delta>-\frac{1}{2}$ and $\beta+\delta=\alpha.$

Substituting this in the expression of ${\tilde{m}}^{\alpha}_{j,R}$ we get that

where $R_{j}=R\sqrt{2^{-j+1}}$.

Notice that in the equation above due to the support of the function $\psi\left(2^{j}\left(1-\frac{|\xi|^{2}}{R^{2}}\right)\right)$ the upper limit of $t$ in the integral is $R_{j}$. Define

and

This yields the following decompostion of the bilinear operator $T^{\alpha}_{j,R}$ associated with the symbol ${\tilde{m}}^{\alpha}_{j,R}(\xi,\eta)$.

where $\alpha=\beta+\delta$.

We will show that the decomposition of the operator as above yields new estimates for the bilinear Bochner-Riesz maximal function. Before going into the proofs of our main results we discuss some useful results concerning the Bochner-Riesz square function.

The Bochner-Riesz square function $G^{\alpha}$ is defined by

where $\widehat{B_{t}^{\alpha}f}(\xi)=\left(1-\frac{|\xi|^{2}}{t^{2}}\right)^{\alpha}_{+}\hat{f}(\xi)$ and $\widehat{\mathcal{K}^{\alpha}_{t}}(\xi)=2(\delta+1)\frac{|\xi|^{2}}{t^{2}}(1-\frac{|\xi|^{2}}{t^{2}})^{\alpha}_{+}.$

The square function $G^{\alpha}$ was introduced by Stein [20]. The $L^{p}$ boundedness

of the square function have been studied extensively in the literature. It has various applications, in particular, it plays an improtant role in the study of maximal Bochner-Riesz functions. We refer the reader to [20, 22, 23, 2, 3, 19, 17, 16, 15] and references there in for details.

Due to the derivative with respect to $t$ the operator $G^{\alpha}$ essentially behaves like the Bochner-Riesz operator $B^{\alpha}$ of order $\alpha$. The Plancherel theorem immediately yields that $G^{\alpha}$ is bounded on $L^{2}({\mathbb{R}}^{n})$ provided $\alpha>-\frac{1}{2}$, see [20]. It is conjectured that for $1<p\leq 2$, the estimate (11) holds if, and only if $\alpha>n(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}.$ Whereas for the range $p>2$ the behaviour of $G^{\alpha}$ is different and it is conjectured that for $p>2$, the estimate (11) holds if, and only if $\alpha>\alpha(p)-\frac{1}{2}.$

The conjecture for the range $1<p\leq 2$ has been resolved. We know that the estimate (11) holds if, and only if $\alpha>n(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}$ for $1<p\leq 2$ and $n\geq 1$, see [22, 23, 17] for detail. Further, in dimensions $n=1,2$, the conjecture has been proved to hold for the range $p\geq 2$ , see [23] and [2] for the case of $n=1$ and $n=2$ respectively. When $n\geq 3$ and $p>2$ the sufficient part of the conjecture is not known completely. There are many interesting developments on the conjecture and we refer to [3, 19, 17, 16, 15] for more detail. The most recent develpoment in this direction is due to  Lee [15], where he proved the following result.

The $L^{p}$ estimates for the square function mentioned as above will be used in order to prove our results. Indeed, we shall need $L^{p}$ boundedness of certain local variants of the square function. We consider the following setting.

For $\psi\in C^{N+1}([\frac{1}{2},2])$ and $0<\nu<\frac{1}{16}$ define

Also, for $\phi\in C^{N}([-1,1])$ and $0<\nu<\frac{1}{16}$ consider the operator

Jeong, Lee and Vargas [Lemma $2.6$ [12]] proved that for any $\epsilon>0$, there exists $N\geq 1$ such that for $\phi\in C^{N}([-1,1])$ the following holds

when $p>\mathfrak{p}_{n}$.

We need the following version of the result stated above.

In Lemma 4.2 we can take the same value of $N$ as in Lemma $2.6$ in [12]. The proof of the lemma above may be completed using Lemma $2.6$ in [12] with minor modifications. We present the proof here for an easy reference and completeness.