<p>We prove weighted weak-type (<i>r</i>,&#xa0;<i>r</i>) estimates for operators satisfying (<i>r</i>,&#xa0;<i>s</i>) limited-range sparse domination of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ell ^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-type. Our results contain improvements for operators satisfying limited-range and square function sparse domination. In the case of operators <i>T</i> satisfying standard sparse form domination such as Calderón-Zygmund operators, we provide a new and simple proof of the sharp bound <Equation ID="Equ1"> <EquationSource Format="TEX">\( \Vert T\Vert _{L^1_w(\textbf{R}^d)\rightarrow L^{1,\infty }_w(\textbf{R}^d)} \lesssim [w]_1(1+\log [w]_{\text {FW}}). \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>T</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msubsup> <mi>L</mi> <mi>w</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="bold">R</mi> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi>L</mi> <mi>w</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>∞</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="bold">R</mi> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≲</mo> <msub> <mrow> <mo stretchy="false">[</mo> <mi>w</mi> <mo stretchy="false">]</mo> </mrow> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mo>log</mo> <msub> <mrow> <mo stretchy="false">[</mo> <mi>w</mi> <mo stretchy="false">]</mo> </mrow> <mtext>FW</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation></p>