<p>This work is devoted to study the relation between regularity and decay for solutions of the two-dimensional modified Zakharov-Kuznetsov equation in the anisotropic weighted Sobolev spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1066_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="350" /> </InlineMediaObject> <EquationSource Format="TEX">\(Z_{s,(r_1,r_2)}:=H^s({\mathbb {R}}^2)\cap L^2((1+|x|^{2r_1}+|y|^{2r_2})dxdy)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>s</mi> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mo>:</mo> <mo>=</mo> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> </mrow> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </msup> <mrow> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>