1. Introduction

Let

\[ \hat{f}(\xi) = \int_{\mathbb{R}^{n}}f(x){\rm e}^{{-}2\pi i x\cdot \xi}\,{\rm d}x, \quad \xi \in \mathbb{R}^{n}, \]

be the Fourier transform of $f \in L^{1}( \mathbb {R}^{n} )$ and let $a_{+} = \max \{a,\,0\}$ denote the positive part of $a \in \mathbb {R}$. Given $\lambda > 0$, the Bochner–Riesz operator $BR_\lambda$ is defined by

\[ \widehat{BR_\lambda f}(\xi) = \left( 1 - |\xi|^{2} \right)^{\lambda}_{+}\hat{f}(\xi), \quad \xi \in \mathbb{R}^{n}. \]

These operators were first introduced by Bochner in [Reference Bochner6] and, since then, they have been widely studied (see [Reference Bourgain and Guth7–Reference Carro, Duoandikoetxea and Lorente9, Reference Christ14, Reference Grafakos22, Reference Kesler and Lacey27, Reference Tao40, Reference Vargas42]). The case $\lambda = 0$ corresponds to the so-called disc multiplier, which is unbounded on $L^{p}(\mathbb {R}^{n})$ if $n \geq 2$ and $p \ne 2$ (see [Reference Fefferman20]).

When $\lambda > {(n - 1)}/2$, it is well known that $BR_\lambda$ is controlled by the Hardy–Littlewood maximal operator $M$, defined for every locally integrable function $f$ (i.e. $f \in L^{1}_{\text {loc}}(\mathbb {R}^{n})$) by

(1.1)\begin{equation} Mf(x) = \sup_{Q \ni x} \frac 1{|Q|} \int_Q |f(y)|\,{\rm d}y, \end{equation}

where the supremum is taken overall cubes $Q \subseteq \mathbb {R}^{n}$ that contains $x \in \mathbb {R}^{n}$. As a consequence, all weighted inequalities for $M$ are also satisfied by $BR_\lambda$.

The value $\lambda = {n - 1}/{2}$ is called the critical index. In this case, Shi and Sun [Reference Shi and Sun38] proved that $BR_{{(n-1)}/2}$ is bounded on $L^{p}(v)$ for every $1 < p < \infty$ and $v \in A_p$. The weak-type inequality for $p = 1$ was first settled by Christ [Reference Christ16], who showed that $BR_{{(n-1)}/2}$ is bounded from $L^{1}(\mathbb {R}^{n})$ to $L^{1,\infty }(\mathbb {R}^{n})$, and the corresponding weighted weak-type inequality was obtained by Vargas in [Reference Vargas42], where she proved that, indeed, $BR_{{(n-1)}/2}$ is bounded from $L^{1}(v)$ to $L^{1,\infty }(v)$ for every $v \in A_1$.

Below the critical index, that is $0 < \lambda < {(n-1)}/2$, $BR_\lambda$ is not bounded on $L^{p}({{\mathbb {R}^{n}}})$ for the whole range $1 < p < \infty$. For instance, in dimension $n=2$, Carleson and Sjölin [Reference Carleson and Sjölin8] proved that $BR_\lambda$ is bounded on $L^{p}({{\mathbb {R}^{n}}})$ if and only if $p > 1$ and

\[ \lambda > \max\left( {2\left|\frac{1}{p} - \frac{1}{2} \right| - \frac{1}{2} , 0 } \right), \]

or equivalently, $0 < \lambda < \frac 12$ and

\[ \frac 4{3 + 2\lambda} < p < \frac 4{1 -2\lambda}. \]

Moreover, Seeger [Reference Seeger37] showed that the corresponding weak-type inequality at the endpoint $BR_\lambda : L^{4/{(3 + 2\lambda )}}({{\mathbb {R}^{n}}}) \rightarrow L^{4/{(3 + 2\lambda )},\, \infty }({{\mathbb {R}^{n}}})$ also holds.

For higher dimensions, it is already well known that $BR_\lambda$ is not bounded on $L^{p}(\mathbb {R}^{n})$ for $p \leq {2n}/{(n + 1 + 2\lambda )}$ or $p \geq {2n}/{(n - 1 - 2\lambda )}$ (see for instance [Reference Duoandikoetxea18, theorem 8.15] or [Reference Grafakos22, proposition 5.2.3]). Furthermore, it was conjectured the following:

Conjecture 1.1 (Bochner–Riesz conjecture)

$BR_\lambda$ is bounded on $L^{p}(\mathbb {R}^{n})$ if $p > 1$ and

\[ \lambda >\lambda(n,p) = \max\left( {n\left|\frac{1}{p} - \frac{1}{2}\right| - \frac{1}{2} , 0 } \right), \]

or equivalently, $0 < \lambda < {(n-1)}/2$ and

\[ \frac{2n}{n + 1 + 2\lambda} < p < \frac{2n}{n - 1 - 2\lambda}. \]

This has an equivalent formulation (see [Reference Lacey, Mena and Reguera29]): let $\chi _{[-1/4, 1/4]} \leq \phi \leq \chi _{[-1/2,1/2]}$ be a Schwartz function, and set $S_\tau$, $\tau > 0$, to be the Fourier multiplier having as a symbol $\phi ((|\xi | - 1)/\tau )$, $\xi \in \mathbb {R}^{n}$.

Conjecture 1.2 For $p > 1$ such that $n\big |\frac 1p - \frac 12 \big | < \frac 12$,

\[ S_\tau: L^{p}({{\mathbb{R}^{n}}}) \rightarrow L^{p}({{\mathbb{R}^{n}}}), \quad 0 < \tau < 1, \]

with constant controlled by $C_\varepsilon \tau ^{-\varepsilon }$ for all $0 < \varepsilon < 1$.

However, the Bochner–Riesz conjecture only has been partially answered and the best results known are currently due to Guo, Oh, Wang, Wu and Zhang (see [Reference Guo, Oh, Wang, Wu and Zhang23]) where the authors used polynomial partitioning. Before them, the best known results followed from the work of [Reference Guth, Hickman and Iliopoulou24] (see also [Reference Wu43]). We thank the referee for this information.

Further, under the condition ${(n - 1)}/{(2(n + 1))} < \lambda < {(n-1)}/2$, the corresponding weak-type inequality at the endpoint $BR_\lambda : L^{{2n}/{(n + 1 + 2\lambda )}} \rightarrow L^{{2n}/{(n + 1 + 2\lambda )},\, \infty }$ was settled by Christ [Reference Christ15] and extended to $\lambda = {(n - 1)}/{(2(n + 1))}$ by Tao [Reference Tao40], while it remains unknown for the range $0 < \lambda < {(n - 1)}/{(2(n + 1))}$. This is the so-called endpoint Bochner–Riesz conjecture (see [Reference Tao41]) and we observe that if it holds for some $\lambda$, then by duality and interpolation the Bochner–Riesz conjecture holds for such $\lambda$ as well.

In two recent papers [Reference Kesler and Lacey27, Reference Lacey, Mena and Reguera29], new weighted estimates for $BR_\lambda$ have been proved using the fact that the Bochner–Riesz operators can be dominated by sparse type operators. As far as we know, these are the best weighted estimates known together with the results for the $(2,2)$-strong type inequality in [Reference Carro, Duoandikoetxea and Lorente9, Reference Christ14]. We summarize them here in a suitable way for the following main results of this paper. To do so, first we need the following notion.

Definition 1.3 Given $0 \leq \alpha,\, \beta \leq 1$ and $1 \leq p < \infty$, let us define the class of weights

\[ A_{p;(\alpha, \beta)} = \{0< v\in L^{1}_{\text{\rm loc}}(\mathbb{R}^{n}) : v = v_0^{\alpha} v_1^{\beta(1-p)}, v_j \in A_1\}, \]

where

(1.2)\begin{equation} v \in A_1 \quad \iff \quad ||v||_{A_1}= \left\Vert \frac{Mv}v\right\Vert_\infty<\infty. \end{equation}

Remark 1.4 $A_{p;(\alpha,\beta )} \subseteq A_p$ (see for instance [Reference Carro, Duoandikoetxea and Lorente9]) where $A_p$ is the Muckenhoupt class of weights (see (2.1)). Besides, we observe that

\[ v\in A_{p;(\alpha, \beta)} \quad\implies\quad v\in A_{1+\beta(p-1)}. \]

Let us start stating the result for dimension $n=2$ where the conjecture has completely been solved.

Further, for $n > 2$ we have the following result.

Now, set $\mathcal {M}(\mathbb {R}^{n})$ to be the set of all measurable functions on $\mathbb {R}^{n}$ and let us consider for a positive and locally integrable function $w$ in $\mathbb {R}^{+}$ (i.e. $w\in L^{1}_{\text {loc}}(\mathbb {R}^{+})$) the primitive $W(t)=\int _0^{t} w(r){\rm d}r$. Given a sublinear operator $T$ defined over the function space

\[ \Lambda^{1}(w) = \left\{ f \in \mathcal{M}(\mathbb{R}^{n}) : ||f||_{\Lambda^{1}(w)} = \int_0^{\infty}f^{*}(t)w(t)\,{\rm d}t < \infty \right\}, \]

we define the class

\[ \mathcal{W}(T)=\Big\{0 \leq w\in L^{1}_{\text{loc}}(\mathbb{R}^{+}): \left\lVert{w}\right\rVert_{\mathcal{W}(T)} = \sup_{f \in \Lambda^{1}(w)}\dfrac{\sup_{y > 0}\, y\, W(\mu_{Tf}(y))}{\int_0^{\infty} f^{*}(t) w(t) {\rm d}t} < \infty\Big\}, \]

where $\mu _f$ and $f^{*}$ are, respectively, the distribution function and the decreasing rearrangement of $f \in \mathcal {M}(\mathbb {R}^{n})$ defined by

(1.3)\begin{equation} \mu_f(t):= |\{|f|>t\}|, \quad f ^{*}(t):=\inf\{y>0:\mu_f(y)\leq t\}, \quad t > 0. \end{equation}

Then, one can immediately see that

\[ w(t)\!=\!t^{{( 1 - n+2\lambda)}/{2n}}\in \mathcal{W}(BR_\lambda)\quad \!\!\implies\!\!\quad \text{ Bochner-{-}Riesz conjecture holds for } BR_\lambda. \]

The original motivation of this paper was to study the class $\mathcal {W}(T)$ not only for the Bochner–Riesz operators but also for other interesting operators in harmonic analysis having in common some properties connected with weighted estimates; namely, for some $1 \leq p < \infty$,

(1.4)\begin{equation} T: L^{p}(v) \longrightarrow L^{p,\infty}(v), \end{equation}

for every positive and locally integrable function $v$ in $\mathbb {R}^{n}$ belonging in $A_{p;(\alpha,\beta )}$ for some $0 \leq \alpha,\,\beta \leq 1$. In particular, and as a consequence of our main results, we will obtain boundedness on weighted Lorentz spaces $\Lambda ^{p}(w)$, $0 < p < \infty$, defined by

\[ \Lambda^{p}(w) = \left\{ f \in \mathcal{M}(\mathbb{R}^{n}) : ||f||_{\Lambda^{p}(w)} = \left(\int_0^{\infty}f^{*}(t)^{p}w(t)\,{\rm d}t \right)^{ 1/p} < \infty \right\}. \]

Weighted type inequalities as in (1.4) are also satisfied by other operators such as the Hörmander Fourier multipliers $m\in M(s,\, l)$ with $l< n$ (see [Reference Kurtz and Wheeden28]) among many others. And this is the starting point of this paper. We want to use the information about weighted estimates on Lebesgue spaces to study the class $\mathcal {W}(T)$ and, as a consequence, to obtain boundedness on weighted Lorentz spaces. To this end, we need to adapt the ideas in [Reference Baena-Miret and Carro4] where the case $\alpha =\beta =1$ was studied and applications to the case $\lambda ={(n-1)}/2$ (for the Bochner–Riesz operator) and the case $l = n$ (for the Hörmander multiplier) were considered. In particular, the following result was proved: if $T$ satisfies that $T:L^{1}(v) \rightarrow L^{1, \infty }(v)$, for every $v \in A_1$, with constant less than or equal to $\varphi (||{v}||_{A_1})$, with $\varphi$ an increasing function on $(0,\, \infty )$, then (see (2.3) for the definition of the class $B_p^{{\mathcal{R}}}$, and definition 2.2 for $B^{*}_\infty$)

(1.5)\begin{equation} B_{1}^{\mathcal{R}}\cap B_{\infty}^{*}\subset \mathcal{W}(T)\quad \text{ with }\quad ||w||_{\mathcal{W}(T)}\leq C_1 ||w||_{B_{1}^{\mathcal{R}}}\varphi\left(C_2 ||w||_{B_{\infty}^{*}} \right) \end{equation}

for some universal constants $C_1,\, C_2>0$.

We write $A \lesssim B$, where $A$, $B$ are nonnegative quantities, if there exists a positive constant $C>0$, independent of $A$ and $B$, such that $A\leq C B$. If $A\lesssim B$ and $B\lesssim A$, then we write $A\approx B$.

Hence, the inclusion in (1.5) can be reinterpreted (see (2.3) and (2.7)) to the existence of some $\varepsilon > 0$ so that if

\[ \frac{W(t)}t \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T)$. This motivates the next theorems that allows us to generalize the results in [Reference Baena-Miret and Carro4] to operators satisfying (1.4). Indeed, (1.5) turns out to be the case $p_0 = \alpha = 1$ of our second main result (see theorem 1.11). Our first main result is the following (see (2.2) for the definition of the class $B_p$, and definition 2.3 for $B^{*}_p$):

Theorem 1.8 Let $1 < p_0 < \infty,$ $0 \leq \alpha \leq 1,$ $0 < \beta \leq 1$ and let $T$ be a sublinear operator such that

\[ T:L^{p_0}(v)\longrightarrow L^{p_0,\infty}(v), \quad \forall v \in A_{p_0;(\alpha, \beta)}, \]

with constant less than or equal to $\varphi (||{v}||_{A_{1+\beta (p_0-1)}}),$ with $\varphi$ an increasing function on $(0,\, \infty )$. Set

\[ p_{+} = \frac{p_0}{1 - \alpha}, \hspace{7mm} p_{-}' = \frac{p_0'}{1-\beta}. \]

Then, $B_{{1}/{p_-}}\cap B_{ {p_+}}^{*} \subset \mathcal {W}(T),$ or equivalently, if there exists some $\varepsilon > 0$ so that

\[ \frac{W(t)}{t^{1/{p_-} - \varepsilon}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{1/{p_+}+\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T)$. Moreover, it holds that $\left \lVert {w}\right \rVert _{\mathcal {W}(T)} \leq \Phi (\left \lVert {w}\right \rVert _{B_{{1}/{p_-}}},\, \left \lVert {w}\right \rVert _{B_{ {p_+}}^{*}}),$ where $\Phi$ is an increasing function in each variable depending on $\varphi$.

By taking $T=M\circ M$ with $M$ the Hardy–Littlewood maximal operator, it is clear that the above result is false with $\varepsilon =0$ since $T$ satisfies the hypothesis of our theorem with $\alpha =1$ and $\beta =1$ but, although $w = 1$ satisfies the hypothesis, $T$ is not of weak type $(1,\,1)$.

Remark 1.9 Recall that a weight $v$ belongs to the reverse Hölder class $RH_q$, $1 < q < \infty$, if for every measurable cube $Q\subset \mathbb {R}^{n}$,

\[ \left(\frac{1}{|Q|}\int_Qv^{q}\right)^{1/q} \lesssim \frac{1}{|Q|}\int_Qv. \]

It is known that $v \in A_p \cap RH_q$ if and only if $v^{q} \in A_{\tilde p}$ with $\tilde p = q(p - 1) + 1$ (see for instance [Reference Johnson and Neugebauer26, (P6)]) so that when $\alpha > 0$,

\[ A_{p_0;(\alpha, \beta)} = A_{{p_0}/{p_-}} \cap RH_{\left( {{p_+}/{p_0}} \right)'}, \]

which keeps some symmetry with the class of weights $B_{1/{p_-}}\cap B_{p_+}^{*}$.

As a consequence of theorem 1.8, we have the following result:

Corollary 1.10 Let $0 < p < \infty$. Under the hypotheses of theorem 1.8, if there exists some $\varepsilon > 0$ so that

\[ \frac{W(t)}{t^{p/{p_-} - \varepsilon}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{ p/{p_+}+\varepsilon}}\approx\uparrow, \]

we have that

\[ T:\Lambda^{p}(w) \longrightarrow \Lambda^{p}(w). \]

Our next main result reads as follows:

Theorem 1.11 Let $1 \leq p_0 < \infty,$ $0 < \alpha \leq 1,$ and let $T$ be a sublinear operator such that

\[ T:L^{p_0,1}(v^{\alpha}) \longrightarrow L^{p_0,\infty}(v^{\alpha}), \quad \forall v \in A_1, \]

with constant less than or equal to $\varphi ( \left \lVert {v}\right \rVert _{A_1}),$ where $\varphi$ is an increasing function. Then, $B_{1/{p_0}}^{{\mathcal{R}}}\cap B_{{p_0}/{(1-\alpha )}}^{*} \subset \mathcal {W}(T),\,$ or equivalently, if there exists some $\varepsilon > 0$ so that

\[ \frac{W(t)}{t^{1/{p_0}}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{{(1 - \alpha)}/{p_0}+\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T)$. Moreover,

\[ \left\lVert{w}\right\rVert_{\mathcal{W}(T)} \lesssim \left\lVert{w}\right\rVert_{B_{1/{p_0}}^{\mathcal{R}}}^{1/{p_0}} \left\{ \begin{array}{@{}cc} \varphi\left( { C \left\lVert{w}\right\rVert_{B^{*}_{{p_0}/{(1-\alpha)}}}^{{p_0}/\alpha} } \right), & 0 < \alpha < 1, \\ \varphi\left( {C \left\lVert{w}\right\rVert_{B_\infty^{*}}} \right), & \alpha = 1. \end{array}\right. \]

Corollary 1.12 Let $0 < p < \infty$ and let us assume the hypotheses of theorem 1.11.

(a) If

\[ \frac{W(t)}{t^{p/{p_0}}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon > 0 :\,\frac{W(t)}{ t^{{ (p(1-\alpha))}/{p_0}+\varepsilon}}\approx\uparrow, \]

then

\[ T:\Lambda^{p,1}(w) \longrightarrow \Lambda^{p,\infty}(w). \]

(b) If

\[ \exists \varepsilon > 0 : \frac{W(t)}{t^{p/{p_0}-\varepsilon}} \approx\downarrow \quad\mbox{and}\quad \,\frac{W(t)}{ t^{{ (p(1-\alpha))}/{p_0}+\varepsilon}}\approx\uparrow, \]

then

\[ T:\Lambda^{p}(w) \longrightarrow \Lambda^{p}(w). \]

The paper is organized as follows. In § 2 we present some definitions, previous results and technical lemmas which shall be used later on, while § 3 contains our main results. The last section (see § 4) will be devoted to obtain estimates in the setting of weighted Lorentz spaces for the Bochner–Riesz operators in the critical range, among others, such as sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$.